Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

Notes on Con v ex Sets, P olytop es, P olyhedra Com binatorial T op ology , V oronoi Diagrams and Delauna y T riangulations Jean Gallier Department of Computer and Informati on Science Univ ersi t y of Penns yl v ania Philadel phi a, P A 19104, U SA e-mail: jean@ cis.upe nn.edu No vem b er 26, 202 4 2 3 Notes on Con v ex Sets, P olytop es, P olyhedra Com binato rial T op ology , V oronoi Diagram s and Delauna y T riangulat ions Jean Gallier Abstract: Some basic mathem atical to ols s uc h as con v ex sets, p o lytopes and com binatorial top ology , are used quite heavily in applied ﬁelds suc h as geometric mo deling, meshing, com- puter vision, medical imaging and rob otics. This repo rt may b e view ed as a tutorial and a set of notes on con v ex sets, p olytopes, p olyhedra, com binatorial top ology , V oronoi Diagra ms and Delaunay T riangulatio ns. It is intend ed for a broad audience of mathematically inclined readers. One of m y (selﬁsh!) motiv ations in writing these notes w as to understand the conce pt of shel ling and ho w it is used to pro v e the famous Euler-P oincar ´ e f orm ula (P oincar ´ e, 1899) and the more recen t Upp er Bound T he or em (McMu llen, 1970) for p olytop es. Another of my motiv ations w as to giv e a “ correct” accoun t of Delaunay triangulations and V oronoi diagrams in terms of (direct and in v erse) stereographic pro jections on to a sphere and pro ve rigorously that the pro jectiv e map that sen ds the (pro jectiv e) sphere to the (pro jectiv e) parab oloid w orks correctly , that is, maps the Delauna y triangulation and V oronoi diagr a m w.r.t. the lifting on to the sphe re to the Delauna y diagram and V oronoi diagrams w.r.t. the traditional lifting on to the parab oloid. Here, the problem is that this map is only w ell deﬁned (total) in pro jectiv e space and we are forced to deﬁne the notion o f con v ex p olyhedron in pro jectiv e space. It turns out that in order t o ac hiev e (ev en partially) the ab o v e goals, I found that it w as necess ary to include quite a bit of back ground material on con v ex sets , p olytop es, p olyhedra and pro jectiv e space s. I hav e included a rather thorough treatmen t of t he equiv alence of V -p olytop es and H -p olytop es and also of the equiv alence of V -p olyhedra and H -p olyhedra, whic h is a bit harder. In particular, the F ourier-Motzkin e l i m ination method (a v ersion of Gaussian elimination for inequalities) is discussed in some detail. I a lso had to include some material on pro jectiv e spaces, pro jectiv e maps and p olar dualit y w.r.t. a nondegenerate quadric in order to deﬁne a suitable notion of “ pro jectiv e p olyhedron” ba sed on cones. T o the b est of our kno wledge, this notion of pro jectiv e p olyhedron is new. W e also believ e that some of our pro ofs establishing the equiv alence of V -p olyhedra and H -p olyhedra are new. Key-w ords: Con v ex sets, p olytop es, p olyhedra, shellings, com binatorial top ology , V oronoi diagrams, Delauna y triangulations. 4 Con ten ts 1 In tro duction 7 1.1 Motiv ations and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Basic P ropert ies of Conv ex Sets 11 2.1 Con v ex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Carath ´ eo dory’s Theore m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 V ertices, Extremal Poin ts and Krein and Milman’ s Theore m . . . . . . . . . 15 2.4 Radon’s and Helly’s Theorem s and Cen terp oints . . . . . . . . . . . . . . . . 18 3 Separation and Supp orting Hyp erplanes 23 3.1 Separation Theorem s and F ark as Lemma . . . . . . . . . . . . . . . . . . . . 23 3.2 Supp orting Hyp erplanes and Mink ow ski’s Prop osition . . . . . . . . . . . . . 34 3.3 P olarit y and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 P olyhedra and Poly top es 41 4.1 P olyhedra, H -Polytopes and V -P olytop es . . . . . . . . . . . . . . . . . . . . 41 4.2 The Equiv alence of H -Polytopes and V -P olytop es . . . . . . . . . . . . . . . 50 4.3 The Equiv alence of H -Polyh edra and V -P olyhedra . . . . . . . . . . . . . . . 51 4.4 F o urier-Motzkin Elimination and Cones . . . . . . . . . . . . . . . . . . . . . 57 5 Pro jectiv e Spaces and Polyhedra , Pola r Dualit y 67 5.1 Pro jectiv e Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Pro jectiv e Polyhe dra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 T angen t Spaces o f Hyp ersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4 Quadrics (Aﬃne, Pro jectiv e) and P olar Dualit y . . . . . . . . . . . . . . . . 86 6 Basics of Com binatorial T opology 95 6.1 Simplicial and P olyhedral Complexes . . . . . . . . . . . . . . . . . . . . . . 95 6.2 Com binatorial and T op o logical Manifolds . . . . . . . . . . . . . . . . . . . . 107 7 Shellings and the Euler-Poinca r´ e F orm ula 111 7.1 Shellings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.2 The Euler-P oincar ´ e F orm ula for P olytop es . . . . . . . . . . . . . . . . . . . 120 5 6 CONTENTS 7.3 Dehn-Sommerville Equations for Simplicial P olytop es . . . . . . . . . . . . . 123 7.4 The Upper Bo und Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8 Diric hlet–V oronoi Diagram s 137 8.1 Diric hlet–V oro noi Diag ra ms . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 T riangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.3 Delauna y T riangulatio ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.4 Delauna y T riangulatio ns and Conv ex Hulls . . . . . . . . . . . . . . . . . . . 14 8 8.5 Stereographic Pro jection and the Space of Spheres . . . . . . . . . . . . . . . 15 1 8.6 Stereographic Pro jection and Delaunay Poly top es . . . . . . . . . . . . . . . 169 8.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Chapter 1 In tro duction 1.1 Motiv ations and Goals F o r the past eigh t y ears or so I hav e b een teac hing a graduate course whose main goal is to expo se stude nts to some fundamen tal concepts of geometry , ke eping in mind their applica- tions to geometric mo delin g, meshing, computer vision, medical imaging, rob otics, etc. The audience has b een primarily computer science studen ts but a fair n um b er of mathematics studen ts and also studen ts fro m other enginee ring disciplines (suc h as Electrical, Systems , Mec hanical and Bio engineering) hav e b een a ttendin g m y classes. In the past three y ears, I hav e b een fo cusing more on con v exit y , p olytop es a nd com binatorial top ology , as concepts and to ols from these areas ha v e b een used increasingly in meshing and also in computational biology and medical imaging. One of my (selﬁsh!) motiv ations w as to understand the con- cept of shel ling and ho w it is used to prov e the famous Euler-P oincar ´ e form ula (P oincar ´ e, 1899) and the more recen t Upp er Bound The or em (McMullen, 1 970) f or p olytop es. Another of m y motiv ations w as to giv e a “correct” accoun t of Delaunay triangulations and V oronoi diagrams in terms of (direct and inv erse) stereographic pro jections on to a sphere and pro ve rigorously that the pro jectiv e map that sends the (pro jectiv e) sphere to the (pro jectiv e) parab oloid works correctly , that is, maps the Delauna y triangulation and V oronoi diagram w.r.t. the lifting on to the sphere to the Delauna y triangulation and V oronoi diagram w.r.t. the lifting on to the parab oloid. Moreo ve r, the pro j ections of these polyhedra on to the h y- p erplane x d +1 = 0, from the sphe re or from the parab oloid, are iden tical. Here, the problem is that this map is only w ell deﬁned (total) in pro jectiv e space and w e are forced to deﬁne the notion of con v ex p olyhedron in pro jectiv e space. It turns out that in order t o ac hiev e (ev en partially) the ab o v e goals, I found that it w as necess ary to include quite a bit of back ground material on con v ex sets , p olytop es, p olyhedra and pro jectiv e space s. I hav e included a rather thorough treatmen t of t he equiv alence of V -p olytop es and H -p olytop es and also of the equiv alence of V -p olyhedra and H -p olyhedra, whic h is a bit harder. In particular, the F ourier-Motzkin e l i m ination method (a v ersion of Gaussian elimination for inequalities) is discussed in some detail. I a lso had to include some material on pro jectiv e spaces, pro jectiv e maps and p olar dualit y w.r.t. a nondegenerate 7 8 CHAPTER 1. INTRODUCTI ON quadric, in order to deﬁne a suitable notion of “pro jectiv e po lyh edron” based on cones. This notion turned out to b e indisp ensible to giv e a correct t r eatmen t of the Delauna y and V oronoi complexes using in v erse stereographic pro jection on to a sphere and to prov e rigo r o usly that the w ell kno wn pro jectiv e map b et w een the sphere and the parab oloid maps the Delauna y triangulation and the V oronoi diagra m w.r.t. the sphere to the more traditional Delauna y triangulation and V oronoi diagram w.r.t. the parab oloid. T o the b est of our kno wledge, this notion of pro jectiv e p olyhedron is new. W e also believ e that some of our pro ofs establishing the equiv alence of V -p olyhed ra and H -p olyhedra are new. Chapter 6 on combi natorial top ology is hardly original. Ho w ev er, most texts cov ering this material are either old fa shion or to o adv anced. Y et, this material is used exten siv ely in meshing and geometric mo delin g. W e tried to giv e a rat her in tuitiv e y et rigorous exposition. W e decided to introduce the terminology c ombinatorial manifold , a no tio n usually referred to as triangulate d manifold . A recurring theme in these notes is the pro cess of “coniﬁcation” (algebraically , “ homoge- nization”), that is, forming a cone from some geometric ob ject. Indeed, “coniﬁcation” turns an ob j ect into a set of lines, and since lines pla y the role of p oin ts in pro jectiv e geome- try , “coniﬁcation” (“homogenization”) is the w a y to “pro jectivize” geometric aﬃne ob jects. Then, these (aﬃne) ob jects a pp ear as “ conic sections” of cones b y h yp erplanes, just the w a y the classical conics (ellipse, h yp erbola, parab o la) app ear as conic sections. It is w orth w arning our readers that con ve xit y and p olytop e theory is deceptiv ely simple. This is a sub ject where most intu itiv e prop ositions fail as so on as the dimension of the space is g reater than 3 (deﬁnitely 4), because our h uman in tuition is not v ery go o d in dimension greater than 3. F urthermore, rigorous pro ofs of seemingly v ery simple facts are often quite complicated and ma y require sophisticated to ols (for example, shellings, for a correct pro of of the Euler-P oincar ´ e form ula). Neve rtheless, readers are urged to strengh ten their geometric in tuition; they should just b e v ery vigilan t! This is a nothe r case where T ate’s famous sa ying is more than p ertinen t: “Reason geometrically , pro v e algebraically .” A t ﬁrst, these notes w ere meant as a compleme nt to Chapter 3 (Prop erties of Con ve x Sets: A Glimpse) of m y b o ok ( Geometric Metho ds and Applications , [2 0 ]). How eve r, they turn out to co v er m uc h more material. F or the reader’s con v enience, I ha v e included Chapter 3 of m y b o ok as part of Chapter 2 of these notes. I also assume some fa miliarit y with aﬃne geometry . The reader ma y wish to revie w the basics of aﬃne geometry . These can b e found in an y standard geometry text (Chapter 2 of Gallier [20] co v ers more than needed for these notes). Most of the material on con v ex sets is taken from Berger [6] ( Geometry I I ). Other relev ant sources includ e Ziegler [43], Gr ¨ un baum [24] V alen tine [41], Barvinok [3], R o ck afellar [32], Bourbaki (T op ological V ector Spaces ) [9] and Lax [26], the last four dealing with aﬃne spaces of inﬁnite dimension. As to p olytopes a nd p olyhedra, “the” clas sic reference is Gr ¨ un baum [24]. Other go o d references include Ziegler [43], Ew ald [18], Crom w ell [14] and Thomas [38]. The recen t b o ok b y Thomas con tains a n excellen t and easy going presen tation of p oly- 1.1. MOTIV A TIONS AND GO ALS 9 top e theory . This b o ok also giv es an in tro duction to the theory of triangulations of p oin t conﬁgurations, including the deﬁnition of secondary p olytopes and state p olytop es, whic h happ en to play a role in certain areas of biology . F or this, a quic k but ve ry eﬃci ent presen- tation of Gr¨ obne r bases is pro vided. W e highly recommend Thomas’s b o ok [38] as further reading. It is also an excellen t preparation for the more adv anced b o ok b y Sturmfels [37]. Ho w ev er, in our opinion, the “bible” on p olytop e theory is without an y conte st, Ziegler [43], a masterly a nd b eautiful piece of mathematics . In fa ct, our Chapter 7 is hea vily inspired b y Chapter 8 of Ziegler. How ev er, the pace of Ziegler’s b o ok is quite brisk and w e hop e that our more p edes trian accoun t will inspire readers to go bac k and read the masters. In a not to o distan t future, I w ould lik e to write ab out constrained D elauna y triangula- tions, a formidable topic, please b e pa tient! I wish to t hank Marcelo Siqueira for catch ing man y ty p os and mistak es and for his man y helpful suggestions regarding the presen tation. At least a third of this man uscript w as written while I w as on sabbatical at INRIA, Sophia An tip olis, in the Ascle pios Pro ject. My deepest thanks to Nic holas Ay ac he and his colleagues (especially Xa vier P ennec and Herv ´ e Delingette) for in viting me to sp end a w onderful and ve ry pro ductiv e year and for making me feel p erfectly at home within the Asclepios Pro ject. 10 CHAPTER 1. INTRODUCTI ON Chapter 2 Basic Prop erties of Con v ex Sets 2.1 Con v ex Se ts Con v ex sets play a v ery imp ortan t role in geometry . In this c hapter we state and pro v e some of the “classics” of con v ex aﬃne geometry: Carath ´ eo dory’s theorem, Radon’s theorem, and Helly’s theorem. These theorems share the prop ert y that they are easy to state, but they are deep, and their pro of, although rat her short, requires a lot of creativit y . Giv en an aﬃne space E , recall that a subset V of E is c onvex if for an y tw o p oin ts a, b ∈ V , w e ha v e c ∈ V for ev ery p oin t c = (1 − λ ) a + λb , with 0 ≤ λ ≤ 1 ( λ ∈ R ). Giv en an y tw o p oin ts a, b , the notation [ a, b ] is often use d to denote the line segmen t b et w een a and b , that is, [ a, b ] = { c ∈ E | c = (1 − λ ) a + λb, 0 ≤ λ ≤ 1 } , and thus a set V is con v ex if [ a, b ] ⊆ V for any t w o p oin ts a, b ∈ V ( a = b is allo w ed). The empt y set is trivially con v ex, ev ery one-p oint set { a } is con v ex, and the en tire aﬃne space E is of course conv ex. It is ob vious that the inters ection of any family (ﬁnite or inﬁnite) of con v ex sets is con v ex. Then, giv en any (nonempt y) subset S of E , there is a smallest con v ex set con taining S denoted b y C ( S ) or con v( S ) and called the c onvex hul l of S (namely , the in tersection of all con v ex sets con taining S ). The aﬃne hul l of a subse t, S , of E is the smalle st aﬃne set con taining S and it will b e denoted by h S i or aﬀ ( S ). A go o d understanding of what C ( S ) is, and g oo d metho ds fo r computing it, are essen tial. First, w e ha v e the follo wing simple but crucial lemma: Lemma 2.1 Given an aﬃne sp ac e  E , − → E , +  , for any family ( a i ) i ∈ I of p oints in E , the set V of c onvex c ombinations P i ∈ I λ i a i (wher e P i ∈ I λ i = 1 and λ i ≥ 0 ) is the c on vex hul l of ( a i ) i ∈ I . Pr o of . If ( a i ) i ∈ I is empty , then V = ∅ , b ecause of the condition P i ∈ I λ i = 1. As in the case of aﬃne com binations, it is easily sho wn b y induction that an y con v ex com bination can b e 11 12 CHAPTER 2. BASIC PR OPER TIES OF CONVEX SETS obtained b y computing con vex com binations of tw o po in t s at a time. As a consequence, if ( a i ) i ∈ I is nonempt y , then the smallest con v ex subs pace con taining ( a i ) i ∈ I m ust con tain the set V of all con v ex combin ations P i ∈ I λ i a i . Th us, it is enough to sho w that V is closed under con v ex combin ations, whic h is immediately v eriﬁed. In view of Lemma 2.1, it is ob vious that an y aﬃne subspace of E is con v ex. Con v ex sets also arise in terms of h yp erplanes. Give n a hyperplane H , if f : E → R is an y noncons tant aﬃne form deﬁning H (i.e., H = Ker f ), w e can deﬁne the t w o subsets H + ( f ) = { a ∈ E | f ( a ) ≥ 0 } and H − ( f ) = { a ∈ E | f ( a ) ≤ 0 } , called (close d ) half-sp ac e s asso cia te d w ith f . Observ e that if λ > 0, then H + ( λf ) = H + ( f ), but if λ < 0, then H + ( λf ) = H − ( f ), and similarly for H − ( λf ). How eve r, the set { H + ( f ) , H − ( f ) } dep ends only on the h yp erplane H , and the c hoice of a sp eciﬁc f deﬁning H amoun ts to the c hoice of o ne of t he tw o half-spaces. F or this reason, w e will a lso sa y that H + ( f ) and H − ( f ) are the close d half-sp ac es asso c iate d with H . C learly , H + ( f ) ∪ H − ( f ) = E and H + ( f ) ∩ H − ( f ) = H . It is immediately v eriﬁed that H + ( f ) and H − ( f ) are con v ex. Bounded con v ex sets arising as the in tersection of a ﬁnite family of half-spaces asso ciated with hyperplanes play a ma jor role in con v ex geometry and topo lo g y (they are called c onvex p ol ytop es ). It is natural to w onder whether Lemma 2.1 can b e sharp ened in tw o directions: (1) Is it p ossible to ha v e a ﬁxed b ound on the num b er of p oin ts in volv ed in the con ve x com binations? (2) Is it nece ssary to consider conv ex com binations of all p oin ts, or is it p ossible to consid er only a subse t with sp ecial properties? The a nsw er is y es in b oth cases. In case 1 , assuming that the aﬃne space E has dimens ion m , Carath ´ eo dory’s theorem asserts that it is enough to consider conv ex com binations of m + 1 p oin ts. F or example, in the plane A 2 , the con v ex hu ll of a set S of p oin ts is the union of all triangles (in terior points includ ed) with v ertices in S . In case 2, the theorem of Krein and Milman asserts that a con v ex set that is also compact is the con v ex h ull of its extremal p oin ts (giv en a conv ex set S , a p oin t a ∈ S is extremal if S − { a } is also con v ex, see Berger [6] or Lang [25]). Next, w e prov e Carath ´ eo dory’s theorem. 2.2 Carath ´ eo dory’s The orem The pro of of Carath ´ eo dory’s theorem is really b eautiful. It pro ceeds b y contradiction and uses a minimalit y argumen t. 2.2. CARA TH ´ EODOR Y’S THEOREM 13 Theorem 2.2 Given any aﬃne sp ac e E of dime n sion m , for any (nonvoid) family S = ( a i ) i ∈ L in E , the c onvex hul l C ( S ) of S is e qual to the set of c onvex c ombinations of families of m + 1 p oints of S . Pr o of . By Lemma 2.1, C ( S ) =  X i ∈ I λ i a i | a i ∈ S, X i ∈ I λ i = 1 , λ i ≥ 0 , I ⊆ L, I ﬁnite  . W e would lik e to pro v e that C ( S ) =  X i ∈ I λ i a i | a i ∈ S, X i ∈ I λ i = 1 , λ i ≥ 0 , I ⊆ L, | I | = m + 1  . W e pro ceed b y contradiction. If the theorem is false, there is some p oin t b ∈ C ( S ) suc h that b can b e expres sed as a con v ex com bination b = P i ∈ I λ i a i , where I ⊆ L is a ﬁnite set of cardinalit y | I | = q with q ≥ m + 2, and b cannot b e expresse d as any con v ex com bination b = P j ∈ J µ j a j of strictly few er than q p oin ts in S , that is, where | J | < q . Suc h a p oint b ∈ C ( S ) is a con v ex com bination b = λ 1 a 1 + · · · + λ q a q , where λ 1 + · · · + λ q = 1 and λ i > 0 (1 ≤ i ≤ q ). W e shall pro v e that b can b e written as a con v ex combin ation of q − 1 o f the a i . Pic k any o rigin O in E . Since there are q > m + 1 p oin ts a 1 , . . . , a q , these p oints a re a ﬃne ly dep enden t, and b y Lemma 2.6.5 from Gallier [20], there is a family ( µ 1 , . . . , µ q ) all scalars not all n ull, suc h that µ 1 + · · · + µ q = 0 and q X i =1 µ i Oa i = 0 . Consider the set T ⊆ R deﬁned by T = { t ∈ R | λ i + tµ i ≥ 0 , µ i 6 = 0 , 1 ≤ i ≤ q } . The set T is nonempt y , sinc e it contains 0. Since P q i =1 µ i = 0 and the µ i are not all n ull, there are some µ h , µ k suc h that µ h < 0 and µ k > 0, whic h implies that T = [ α, β ], where α = max 1 ≤ i ≤ q {− λ i /µ i | µ i > 0 } and β = min 1 ≤ i ≤ q {− λ i /µ i | µ i < 0 } ( T is the in tersection of the closed half-spaces { t ∈ R | λ i + tµ i ≥ 0 , µ i 6 = 0 } ). Observ e that α < 0 < β , since λ i > 0 for all i = 1 , . . . , q . W e claim that there is some j (1 ≤ j ≤ q ) suc h that λ j + α µ j = 0 . 14 CHAPTER 2. BASIC PR OPER TIES OF CONVEX SETS Indeed, since α = max 1 ≤ i ≤ q {− λ i /µ i | µ i > 0 } , as the set on the righ t hand side is ﬁnite, the maxim um is ac hiev ed and there is some index j so that α = − λ j /µ j . If j is some index suc h that λ j + α µ j = 0, sinc e P q i =1 µ i Oa i = 0, w e ha v e b = q X i =1 λ i a i = O + q X i =1 λ i Oa i + 0 , = O + q X i =1 λ i Oa i + α  q X i =1 µ i Oa i  , = O + q X i =1 ( λ i + α µ i ) Oa i , = q X i =1 ( λ i + α µ i ) a i , = q X i =1 , i 6 = j ( λ i + α µ i ) a i , since λ j + α µ j = 0. Since P q i =1 µ i = 0, P q i =1 λ i = 1, and λ j + α µ j = 0, w e hav e q X i =1 , i 6 = j λ i + α µ i = 1 , and since λ i + αµ i ≥ 0 for i = 1 , . . . , q , the ab o v e sho ws that b can b e expre ssed as a conv ex com bination of q − 1 p oin ts from S . Ho w ev er, this con tradicts the assumption that b cannot b e expressed as a con v ex com bination of strictly few er than q p oin ts fro m S , and the theorem is pro v ed. If S is a ﬁnite ( o f inﬁnite) set of p oin ts in the aﬃne plane A 2 , Theorem 2.2 conﬁrms our in tuition that C ( S ) is the union of triangles (including interior p oin ts) whose v ertices b elong to S . Similarly , the con v ex h ull of a set S of p oin ts in A 3 is the union of tetrahedra (including in terior p o in t s) whose v ertices b elong to S . W e get the f eeling that triangulations pla y a crucial role, whic h is of course true! No w that we hav e giv en an answ er to the ﬁrst question p osed at the end of Section 2.1 w e giv e an answ er to the sec ond question. 2.3. VER TICES, EXTREMAL POINTS AND KREIN AND MILMAN’S THEOREM 15 2.3 V ertice s, E xtre mal P oints and Krein and Milman’s Theorem First, w e deﬁne the notions o f separation and of separating h yperplanes. F or this, recall the deﬁnition of the closed (or op en) half– spaces determined b y a hyperplane. Giv en a hyperplane H , if f : E → R is any nonconstan t aﬃne f o rm deﬁning H (i.e., H = Ker f ), we deﬁne the clos e d h alf-sp ac es asso ciate d with f b y H + ( f ) = { a ∈ E | f ( a ) ≥ 0 } , H − ( f ) = { a ∈ E | f ( a ) ≤ 0 } . Observ e that if λ > 0, then H + ( λf ) = H + ( f ), but if λ < 0, then H + ( λf ) = H − ( f ), and similarly for H − ( λf ). Th us, the set { H + ( f ) , H − ( f ) } dep ends only on the hyperplane, H , and the c hoice of a sp eciﬁ c f deﬁning H amoun ts to the choice of one of the t w o half-spaces. W e also deﬁne the op en half–sp ac es as s o ciate d with f as the t wo sets ◦ H + ( f ) = { a ∈ E | f ( a ) > 0 } , ◦ H − ( f ) = { a ∈ E | f ( a ) < 0 } . The set { ◦ H + ( f ) , ◦ H − ( f ) } only dep end s on the hy p erplane H . Clearly , w e hav e ◦ H + ( f ) = H + ( f ) − H and ◦ H − ( f ) = H − ( f ) − H . Deﬁnition 2.1 Giv en an aﬃne space, X , and t w o nonempt y sub sets, A and B , of X , w e sa y that a hyperplane H sep ar ates (r esp. strictly se p ar ates) A and B if A is in one and B is in the other of the t w o half–spaces (resp. op en half–spaces) determin ed b y H . The sp ecial case of separation where A is conv ex and B = { a } , for some p oin t, a , in A , is of particular imp ortance. Deﬁnition 2.2 Let X b e an aﬃne space and let A b e any nonempt y subset of X . A sup- p orting hyp erplane of A is an y h yp erplane, H , containing some p oin t, a , o f A , and separating { a } and A . W e sa y that H is a supp orting hyp erplane of A at a . Observ e that if H is a supp orting h yp erplane of A at a , then w e m ust ha v e a ∈ ∂ A . Otherwise, there w ould b e some op en ball B ( a, ǫ ) of cen ter a containe d in A and so there w ould b e p oin ts of A (in B ( a, ǫ )) in bo th half- space s determine d b y H , contradicting the fact that H is a supp orting h yp erplane of A at a . F urthermore, H ∩ ◦ A = ∅ . One should exp erimen t with v a rio us pictures a nd realize that suppor t ing hy p erplanes at a p oin t may not exist (for example, if A is not conv ex), ma y not b e unique, and may hav e sev eral distinct supp orting p oin ts! Next, w e need to deﬁne v a rious types of b oundary p oin ts of closed con v ex sets . 16 CHAPTER 2. BASIC PR OPER TIES OF CONVEX SETS Figure 2.1: Examples of supp orting h yp erplanes Deﬁnition 2.3 Let X b e an aﬃne space of dimension d . F or any nonempt y closed and con v ex subset, A , of dimension d , a p oin t a ∈ ∂ A has or der k ( a ) if the inters ection of all the supp orting h yp erplanes of A at a is an aﬃne subspace of dimens ion k ( a ). W e say that a ∈ ∂ A is a vertex if k ( a ) = 0; we sa y that a is smo oth if k ( a ) = d − 1, i.e., if the supporting h yp erplane at a is unique. A vertex is a b oundary p oin t, a , suc h that there are d independen t suppor ting hyperplanes at a . A d -simplex has b oundary p oin ts of order 0 , 1 , . . . , d − 1. The follo wing prop osition is sho wn in Berger [6] (Prop osition 11.6.2): Prop osition 2.3 The set of vertic es of a c lose d and c onvex subset is c ountable. Another imp ortan t concept is that of an extremal p oin t. Deﬁnition 2.4 Let X b e a n aﬃne space. F or any nonempt y conv ex subset, A , a p oin t a ∈ ∂ A is extr emal (or extr eme ) if A { a } is still con v ex. It is fairly ob vious that a p o in t a ∈ ∂ A is extremal if it do es not b elong to an y close d non trivial line segmen t [ x, y ] ⊆ A ( x 6 = y ). Observ e that a v ertex is extremal, but the conv erse is false. F o r example, in Figure 2.2, all the p oin ts on the arc o f para b ola, including v 1 and v 2 , are extreme p oin ts. Ho w ev er, only v 1 and v 2 are v ertices. Also, if dim X ≥ 3, the set of extremal p oin ts of a compact con v ex ma y not b e closed. Actually , it is not at a ll obvi ous that a nonempt y compact con v ex set p ossesses extremal p oin ts. In fact, a stronger results holds (Krein and Milman’s theorem). In preparation fo r the pro of of this impo r t a n t theorem, observ e that an y compact (non trivial) in terv al of A 1 has t w o extremal p oints , its tw o endp o in t s. W e need the follo wing lemma: 2.3. VER TICES, EXTREMAL POINTS AND KREIN AND MILMAN’S THEOREM 17 v 1 v 2 Figure 2.2: Examples of v ertices and extreme p oin ts Lemma 2.4 L et X b e an aﬃne sp ac e of dimens i o n n , and let A b e a nonempty c omp act and c onvex set. Then, A = C ( ∂ A ) , i.e., A is e qual to the c onvex hul l of its b oundary. Pr o of . Pic k an y a in A , and consider an y line, D , through a . Then, D ∩ A is closed and con v ex. Ho w ev er, since A is compact, it follow s that D ∩ A is a closed interv al [ u, v ] con taining a , and u, v ∈ ∂ A . Therefore, a ∈ C ( ∂ A ), as desired. The followin g imp ortan t theorem sho ws that only extremal points matter as far as de- termining a compact and conv ex subset from its b oundary . The pro of of Theorem 2.5 mak es use of a prop osition due to Mink owsk i (Prop osition 3.17) whic h will b e pro v ed in Section 3.2. Theorem 2.5 (Kr ein and Milman, 1940) L et X b e an aﬃne sp ac e of dimensio n n . Every c om p act and c o nvex nonempty subset, A , is e qual to the c onvex hul l of its set of extr emal p oi n ts. Pr o of . Denote the set of extremal p oin ts of A b y Extrem( A ). W e pro ceed b y induction on d = dim X . When d = 1, the conv ex and compact subs et A m ust b e a closed interv al [ u, v ], or a single p oin t. In either cases, the theorem holds trivially . Now, a ssume d ≥ 2, and assume that the theorem holds for d − 1. It is easily v eriﬁed that Extrem( A ∩ H ) = (Extrem( A )) ∩ H , for ev ery supporting h yp erplane H of A (suc h h yperplanes exist, b y Mink o wski’s prop osition (Prop osition 3.17)). Observ e that Lemma 2.4 implies that if w e can pro v e that ∂ A ⊆ C (Extrem( A )) , then, since A = C ( ∂ A ), w e will hav e establis hed that A = C (Extrem( A )) . Let a ∈ ∂ A , and let H b e a supp orting h yp erplane of A at a (whic h exists, b y Mink ows ki’s prop osition). Now, A ∩ H is conv ex and H has dimension d − 1 , and by the induction h yp othesis , w e hav e A ∩ H = C (Extrem( A ∩ H )) . 18 CHAPTER 2. BASIC PR OPER TIES OF CONVEX SETS Ho w ev er, C (Extrem( A ∩ H )) = C ((Extrem ( A )) ∩ H ) = C (Extrem( A )) ∩ H ⊆ C (Extrem( A )) , and so, a ∈ A ∩ H ⊆ C (Extrem( A )). Therefore, w e prov ed that ∂ A ⊆ C (Extrem( A )) , from whic h w e deduce that A = C (Extrem( A )), as explained earlier. Remark: Observ e that Krein and Milman’s theorem implies that a n y nonempt y compact and con v ex set has a nonempt y subset of extremal p oin ts. This is in tuitiv ely ob vious, but hard to pro v e! Krein and Milman’s theorem also applies to inﬁnite dimens ional aﬃne spaces, pro vided that they are lo cally con v ex, see V alentine [41], Chapter 11, Bourbaki [9], Chapter I I, Barvinok [3], Chapter 3, or Lax [26], Chapter 13. W e conclude this c ha pter with three other classics o f conv ex geometry . 2.4 Radon’s and Helly’s Theorems and Ce n te rp oin ts W e b egin with R a d on ’s the o r em . Theorem 2.6 Given any aﬃn e sp ac e E of dimension m , for eve ry subset X of E , if X has at le ast m + 2 p oints, then ther e is a p artition of X into two nonempty disjoint subsets X 1 and X 2 such that the c onvex hul ls of X 1 and X 2 have a non e m pty interse ction. Pr o of . Pic k some origin O in E . W rite X = ( x i ) i ∈ L for some index set L (we can let L = X ). Since b y assumption | X | ≥ m + 2 where m = dim( E ), X is aﬃnely dep enden t, and b y Lemma 2.6.5 from Gallier [20 ], there is a family ( µ k ) k ∈ L (of ﬁnite supp ort) of scalars, not all n ull, suc h that X k ∈ L µ k = 0 a nd X k ∈ L µ k Ox k = 0 . Since P k ∈ L µ k = 0, the µ k are not all n ull, and ( µ k ) k ∈ L has ﬁnite support, the sets I = { i ∈ L | µ i > 0 } a nd J = { j ∈ L | µ j < 0 } are nonempt y , ﬁnite, a nd ob viously disjoin t. Let X 1 = { x i ∈ X | µ i > 0 } a nd X 2 = { x i ∈ X | µ i ≤ 0 } . Again, since the µ k are not all null and P k ∈ L µ k = 0, the sets X 1 and X 2 are nonempt y , and ob viously X 1 ∩ X 2 = ∅ a nd X 1 ∪ X 2 = X . 2.4. RADON’S AND HELL Y’S THEOREMS AND CENTERPOINT S 19 F urthermore, the deﬁnition of I and J implies that ( x i ) i ∈ I ⊆ X 1 and ( x j ) j ∈ J ⊆ X 2 . It remains to pro v e tha t C ( X 1 ) ∩ C ( X 2 ) 6 = ∅ . The deﬁnition of I and J implies that X k ∈ L µ k Ox k = 0 can b e written as X i ∈ I µ i Ox i + X j ∈ J µ j Ox j = 0 , that is, as X i ∈ I µ i Ox i = X j ∈ J − µ j Ox j , where X i ∈ I µ i = X j ∈ J − µ j = µ, with µ > 0. Th us, we hav e X i ∈ I µ i µ Ox i = X j ∈ J − µ j µ Ox j , with X i ∈ I µ i µ = X j ∈ J − µ j µ = 1 , pro ving that P i ∈ I ( µ i /µ ) x i ∈ C ( X 1 ) and P j ∈ J − ( µ j /µ ) x j ∈ C ( X 2 ) are iden tical, and th us that C ( X 1 ) ∩ C ( X 2 ) 6 = ∅ . Next, w e pro ve a v ersion of Hel ly’s the or em . Theorem 2.7 Given any aﬃne sp ac e E of dim e nsion m , for eve ry family { K 1 , . . . , K n } of n c on v ex subsets of E , if n ≥ m + 2 and the in terse ction T i ∈ I K i of any m + 1 of the K i is nonempty ( w her e I ⊆ { 1 , . . . , n } , | I | = m + 1 ), then T n i =1 K i is nonempty. Pr o of . The pro of is b y induction on n ≥ m + 1 and uses Radon’s theorem in the induction step. F or n = m + 1, the assumption of the theorem is that the inters ection of an y family of m + 1 of the K i ’s is nonempt y , and the theorem holds trivially . Next, let L = { 1 , 2 , . . . , n + 1 } , where n + 1 ≥ m + 2. By the induction h yp othesis, C i = T j ∈ ( L −{ i } ) K j is nonempty for ev ery i ∈ L . W e claim that C i ∩ C j 6 = ∅ for some i 6 = j . If so, a s C i ∩ C j = T n +1 k =1 K k , w e are done. So, let us assume that the C i ’s are pairwise disjoin t. Then, w e can pic k a set X = { a 1 , . . . , a n +1 } suc h that a i ∈ C i , for ev ery i ∈ L . By Radon’s Theorem, there are tw o nonem pty disjoin t sets X 1 , X 2 ⊆ X such that X = X 1 ∪ X 2 and C ( X 1 ) ∩ C ( X 2 ) 6 = ∅ . Ho w ev er, X 1 ⊆ K j for ev ery j with a j / ∈ X 1 . This is b ecause a j / ∈ K j for ev ery j , and so, w e get X 1 ⊆ \ a j / ∈ X 1 K j . 20 CHAPTER 2. BASIC PR OPER TIES OF CONVEX SETS Symetrically , we also hav e X 2 ⊆ \ a j / ∈ X 2 K j . Since the K j ’s are con v ex and   \ a j / ∈ X 1 K j   ∩   \ a j / ∈ X 2 K j   = n +1 \ i =1 K i , it follo ws that C ( X 1 ) ∩ C ( X 2 ) ⊆ T n +1 i =1 K i , so that T n +1 i =1 K i is nonempt y , con tradicting the fact that C i ∩ C j = ∅ for all i 6 = j . A more general v ersion of Helly’s theorem is pro v ed in Berger [6]. An am using corollary of Helly’s theorem is the follo wing result: Consider n ≥ 4 parallel line segmen ts in the aﬃne plane A 2 . If ev ery three of these line segmen ts meet a line, then all of these line segm ents meet a common line. W e conclude this c hapter with a nice application of Helly’s Theorem to the existence of cente rp oin ts. Cen terp oin ts generalize the notion of med ian to higher dimensions. Recall that if w e ha v e a set of n data p oin ts, S = { a 1 , . . . , a n } , on the real line, a me dian f or S is a p oin t, x , suc h that at least n/ 2 of the p oin ts in S b elong to b oth in terv als [ x, ∞ ) and ( −∞ , x ]. Giv en any hyperplane, H , recall that the closed half- spaces determined by H are denoted H + and H − and that H ⊆ H + and H ⊆ H − . W e let ◦ H + = H + − H a nd ◦ H − = H − − H b e the op e n half-sp ac e s determined b y H . Deﬁnition 2.5 Let S = { a 1 , . . . , a n } b e a set of n p oin ts in A d . A p oin t, c ∈ A d , is a c en terp oint of S iﬀ for ev ery h yp erplane , H , whenev er the closed half-space H + (resp. H − ) con tains c , then H + (resp. H − ) con tains at least n d +1 p oin ts from S . So, for d = 2, for eac h line, D , if the closed half- plane D + (resp. D − ) con tains c , then D + (resp. D − ) con tains at least a third of the p oints from S . F or d = 3, for eac h plane, H , if the closed half-space H + (resp. H − ) con tains c , then H + (resp. H − ) con tains at least a fourth of the p oin ts from S , etc. Observ e that a p oin t, c ∈ A d , is a cen terp oin t of S iﬀ c belongs to ev ery op en half-space, ◦ H + (resp. ◦ H − ) con taining at least dn d +1 + 1 p oin ts fro m S . Indeed, if c is a cen terp oin t o f S and H is any hyperplane suc h that ◦ H + (resp. ◦ H − ) con tains a t least dn d +1 + 1 points from S , then ◦ H + (resp. ◦ H − ) mu st con ta in c as otherwise, the closed half-space, H − (resp. H + ) w ould con tain c and a t most n − dn d +1 − 1 = n d +1 − 1 p oin ts from S , a con tradiction. Conv ersely , assume that c b elongs to eve ry open half-space, 2.4. RADON’S AND HELL Y’S THEOREMS AND CENTERPOINT S 21 ◦ H + (resp. ◦ H − ) con taining at least dn d +1 + 1 p oin ts from S . Then, for any h yperplane, H , if c ∈ H + (resp. c ∈ H − ) but H + con tains at most n d +1 − 1 p oin ts from S , then the op en half-space, ◦ H − (resp. ◦ H + ) w ould contain at least n − n d +1 + 1 = dn d +1 + 1 p oin ts from S but not c , a con tradiction. W e are now ready to pro v e the existenc e of cen terp oin ts. Theorem 2.8 Every ﬁnite set, S = { a 1 , . . . , a n } , of n p o i nts in A d has s o me c enterp oint. Pr o of . W e will use the second c haracterization of cen terp oin ts inv olving op en half - space s con taining at least dn d +1 + 1 p oin ts. Consider the family of sets, C =  con v( S ∩ ◦ H + ) | ( ∃ H )  | S ∩ ◦ H + | > dn d + 1  ∪  con v( S ∩ ◦ H − ) | ( ∃ H )  | S ∩ ◦ H − | > dn d + 1  , where H is a h yp erplane. As S is ﬁnite, C consists o f a ﬁnite n um b er of con v ex sets, sa y { C 1 , . . . , C m } . If w e pro v e that T m i =1 C i 6 = ∅ w e are done, b ecause T m i =1 C i is the set of cen t erp oints of S . First, w e pro v e b y induction on k (with 1 ≤ k ≤ d + 1), that any in tersection of k of the C i ’s has at least ( d +1 − k ) n d +1 + k elemen ts from S . F or k = 1 , this holds by deﬁnition of the C i ’s. Next, consider the inters ection of k + 1 ≤ d + 1 of the C i ’s, sa y C i 1 ∩ · · · ∩ C i k ∩ C i k +1 . Let A = S ∩ ( C i 1 ∩ · · · ∩ C i k ∩ C i k +1 ) B = S ∩ ( C i 1 ∩ · · · ∩ C i k ) C = S ∩ C i k +1 . Note that A = B ∩ C . By the induction h yp othesis, B con tains at least ( d +1 − k ) n d +1 + k elemen ts from S . As C con tains at least dn d +1 + 1 p oin ts from S , and as | B ∪ C | = | B | + | C | − | B ∩ C | = | B | + | C | − | A | and | B ∪ C | ≤ n , w e get n ≥ | B | + | C | − | A | , that is, | A | ≥ | B | + | C | − n. It follo ws that | A | ≥ ( d + 1 − k ) n d + 1 + k + dn d + 1 + 1 − n 22 CHAPTER 2. BASIC PR OPER TIES OF CONVEX SETS that is, | A | ≥ ( d + 1 − k ) n + dn − ( d + 1) n d + 1 + k + 1 = ( d + 1 − ( k + 1)) n d + 1 + k + 1 , establishing the induction h yp othesis. No w, if m ≤ d + 1, the ab o v e claim for k = m sho ws that T m i =1 C i 6 = ∅ and w e ar e done. If m ≥ d + 2, the ab o v e claim for k = d + 1 sho ws that a n y in tersection of d + 1 of the C i ’s is nonempt y . Conse quen tly , the conditions for applyin g Helly ’s Theorem are satisﬁed and therefore, m \ i =1 C i 6 = ∅ . Ho w ev er, T m i =1 C i is the set of cen terp oin ts of S and w e are done. Remark: The ab ov e pro of actually sho ws that the set of cente rp oin ts of S is a con v ex set. In fact, it is a ﬁnite in tersection of con v ex h ulls of ﬁnitely man y p oin ts, so it is the con v ex h ull of ﬁnitely man y p oints , in other words, a polytop e. Jadha v and Muk hopadhy ay hav e giv en a linear-time algorithm for computing a cen ter- p oin t of a ﬁnite set of points in the plane. F or d ≥ 3 , it app ears that t he b est that can b e done (using linear programming) is O ( n d ). Ho w ev er, there are go o d approxi mation algo- rithms (Clarkson, Eppstein, Miller, Sturtiv a n t and T eng) and in E 3 there is a near quadratic algorithm (Agarw al, Sharir and W elzl). Chapter 3 Separation and Supp orting Hyp erplanes 3.1 Separation Theore ms and F ark as Lemma It seems in tuitiv ely rather ob vious that if A a nd B are tw o nonempt y disjoin t con v ex sets in A 2 , then there is a line, H , separating them, in the sense that A and B b elong to the tw o (disjoin t) op en half–planes determined b y H . Ho w ev er, this is not alwa ys true! F or example, this fails if b oth A and B are closed and un b ounded (ﬁnd a n example). Nev ertheless, the result is true if both A and B are op en, or if the notion of separation is w eak ened a little bit. The k ey result, from whic h most separation r esults f o llo w, is a geometric v ersion of the Hahn-Banach the or em . In the sequel, w e restrict our attention to real aﬃne spaces of ﬁnite dimension. Then, if X is an aﬃne space of dimension d , there is an aﬃne bijection f betw een X and A d . No w, A d is a topo logical space, under the usual to p ology on R d (in fact, A d is a metric space). Recall that if a = ( a 1 , . . . , a d ) and b = ( b 1 , . . . , b d ) are any t w o p oin ts in A d , their Euclidean distance , d ( a, b ), is give n by d ( a, b ) = p ( b 1 − a 1 ) 2 + · · · + ( b d − a d ) 2 , whic h is also the norm , k ab k , of the v ector ab a nd that for a n y ǫ > 0, the op en b al l of c enter a and r adius ǫ , B ( a, ǫ ), is giv en by B ( a, ǫ ) = { b ∈ A d | d ( a, b ) < ǫ } . A subset U ⊆ A d is op en (in the norm top olo gy ) if either U is empt y or f or ev ery po int, a ∈ U , there is some (small) op en ball, B ( a, ǫ ), con tained in U . A subse t C ⊆ A d is cl o se d iﬀ A d − C is op en. F or example, the close d b al ls , B ( a, ǫ ), where B ( a, ǫ ) = { b ∈ A d | d ( a, b ) ≤ ǫ } , 23 24 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES are closed . A subs et W ⊆ A d is b ounde d iﬀ there is some ball (op en or closed), B , so that W ⊆ B . A subs et W ⊆ A d is c omp act iﬀ ev ery family , { U i } i ∈ I , that is an op en co v er of W (whic h means that W = S i ∈ I ( W ∩ U i ), with eac h U i an op en set) p ossesses a ﬁnite sub co v er (whic h means that there is a ﬁnite subse t, F ⊆ I , so that W = S i ∈ F ( W ∩ U i )). In A d , it can b e sho wn that a subset W is compact iﬀ W is closed and b ounded. Given a function, f : A m → A n , we sa y that f is c ontinuous if f − 1 ( V ) is open in A m whenev er V is o pen in A n . If f : A m → A n is a con tin uous function, although it is generally false that f ( U ) is op en if U ⊆ A m is op en, it is easily che ck ed that f ( K ) is compact if K ⊆ A m is compact. An aﬃne space X o f dimension d b ecomes a top ological space if w e g iv e it the t o polog y for whic h the op en subse ts are of the form f − 1 ( U ), whe re U is an y op en subset of A d and f : X → A d is an aﬃne bijection. Giv en an y subset, A , of a top o logical space , X , t he smallest closed set con taining A is denoted by A , and is called the closur e or adher enc e of A . A subset, A , o f X , is dense in X if A = X . The largest op en set con tained in A is denoted b y ◦ A , and is called the interior of A . The set, F r A = A ∩ X − A , is called the b oundary (or fr o ntier ) of A . W e also denote the b oundary of A b y ∂ A . In order to prov e the Hahn-Banac h theorem, w e will need tw o lemm as. G iv en an y t w o distinct p oin ts x, y ∈ X , w e let ] x, y [ = { (1 − λ ) x + λy ∈ X | 0 < λ < 1 } . Our ﬁrst lemma (Lemma 3 .1) is in tuitiv ely quite ob vious so the reader migh t b e puzzled b y the length of its pro of. Ho w ev er, after pro p osing sev eral wrong pro ofs, w e realized that its pro of is more subtle than it might app ear. The pro of b elo w is due to V alen tine [41]. See if y ou can ﬁnd a shorter (and correct) pro of ! Lemma 3.1 L et S b e a nonempty c onvex se t and let x ∈ ◦ S and y ∈ S . Then, w e have ] x, y [ ⊆ ◦ S . Pr o of . Let z ∈ ] x, y [ , that is, z = (1 − λ ) x + λy , with 0 < λ < 1. Since x ∈ ◦ S , w e can ﬁnd some op en subset, U , con tained in S so that x ∈ U . It is easy to c hec k that the cen tral magniﬁcation of cen ter z , H z , λ − 1 λ , maps x to y . Then, H z , λ − 1 λ ( U ) is an op en subset con taining y and as y ∈ S , w e hav e H z , λ − 1 λ ( U ) ∩ S 6 = ∅ . Let v ∈ H z , λ − 1 λ ( U ) ∩ S b e a p oin t of S in this in tersection. Now , there is a unique p oin t, u ∈ U ⊆ S , suc h that H z , λ − 1 λ ( u ) = v a nd, a s S is con v ex, we deduce that z = (1 − λ ) u + λv ∈ S . Since U is op en, the set (1 − λ ) U + λv = { (1 − λ ) w + λv | w ∈ U } ⊆ S is also op en and z ∈ (1 − λ ) U + λv , whic h show s that z ∈ ◦ S . 3.1. SEP ARA TION THEOREMS AND F ARKAS LEMMA 25 Corollary 3.2 If S is c onvex, then ◦ S is also c onvex, and w e have ◦ S = ◦ S . F urthermor e, if ◦ S 6 = ∅ , then S = ◦ S .  Bew are that if S is a closed set, then the con v ex h ull, con v( S ), of S is not necessarily closed! (F ind a coun ter-example.) Ho w ev er, it can b e sho wn that if S is compact, then con v( S ) is also compact and thu s, closed . There is a simple criterion to test whether a con ve x set has an empty in terior, based on the notion of dimension of a con v ex set. Deﬁnition 3.1 The dimension of a nonempt y con v ex subset, S , of X , denoted b y dim S , is the dimension of the smalles t aﬃne subset, h S i , con taining S . Prop osition 3.3 A nonempty c onvex s e t S has a nonempty interior iﬀ dim S = dim X . Pr o of . Let d = dim X . First, assume that ◦ S 6 = ∅ . Then, S con tains some op en ball of cen ter a 0 , and in it, w e can ﬁnd a frame ( a 0 , a 1 , . . . , a d ) for X . Thus , dim S = dim X . Con v ersely , let ( a 0 , a 1 , . . . , a d ) b e a frame of X , with a i ∈ S , for i = 0 , . . . , d . Then, w e hav e a 0 + · · · + a d d + 1 ∈ ◦ S , and ◦ S is nonempt y .  Prop osition 3.3 is f alse in inﬁnite dime nsion. W e leav e the follo wing prop ert y as an exercise: Prop osition 3.4 If S is c onvex, then S is also c on v ex. One can also easily prov e that con v exit y is preserv ed under direct image and in v erse image b y an aﬃne map. The next lemma, whic h seem s in tuitiv ely ob vious, is the core of the pro of of the Hahn- Banac h theorem. This is the case where the aﬃne space has dimension tw o. First, w e need to deﬁne what is a con v ex cone. Deﬁnition 3.2 A con v ex set, C , is a c onvex c one w ith vertex x if C is in v ariant under all cen tral magniﬁcations, H x,λ , of cen ter x and ratio λ , with λ > 0 (i.e., H x,λ ( C ) = C ). Giv en a con v ex set, S , and a p oint, x / ∈ S , we can deﬁne cone x ( S ) = [ λ> 0 H x,λ ( S ) . It is easy to c hec k that this is a con v ex cone. 26 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES B O C x L Figure 3.1: Hahn-Banach Theorem in the plane (Lemma 3.5) Lemma 3.5 L et B b e a nonempty op en and c onvex subset of A 2 , and let O b e a p oint of A 2 so that O / ∈ B . Then, ther e is some lin e , L , thr ough O , so that L ∩ B = ∅ . Pr o of . Deﬁne the con v ex cone C = cone O ( B ). As B is op en, it is easy to c hec k that eac h H O ,λ ( B ) is op en and since C is the union of the H O ,λ ( B ) (for λ > 0), whic h are op en, C itself is op en. Also, O / ∈ C . W e claim that a least one p oin t, x , of the b oundary , ∂ C , of C , is distinct from O . Otherwise, ∂ C = { O } and w e claim that C = A 2 − { O } , whic h is not con v ex, a contradiction. Indeed, as C is con v ex it is connec ted, A 2 − { O } itself is connected and C ⊆ A 2 − { O } . If C 6 = A 2 − { O } , pic k some p oin t a 6 = O in A 2 − C and some p oin t c ∈ C . No w, a basic prop ert y of connectivi ty asserts that ev ery con tinuous path fro m a (in the exterior of C ) to c (in the in terior of C ) mus t in tersect the b oundary of C , namely , { O } . Ho w ev er, there are plen ty of paths from a to c that av oid O , a con tradiction. Therefore, C = A 2 − { O } . Since C is op en and x ∈ ∂ C , w e hav e x / ∈ C . F urthermore, w e claim that y = 2 O − x (the symmetric of x w.r.t. O ) do es not b elong to C either. Otherwise, we w ould ha v e y ∈ ◦ C = C and x ∈ C , and by Lemma 3.1, w e w ould g et O ∈ C , a con tradiction. Therefore, the line through O and x miss es C en tirely (since C is a cone), and th us, B ⊆ C . Finally , we come to the Hahn-Banach theorem. Theorem 3.6 (Hahn-Banach The or em, ge ometric form) L et X b e a (ﬁnite-dimen s ional) aﬃne sp ac e, A b e a nonempty op en and c onvex subset of X and L b e an aﬃne subsp ac e of X so that A ∩ L = ∅ . Then, ther e is some hyp erplane, H , c ontainin g L , that i s disjoint fr om A . Pr o of . The case where dim X = 1 is trivial. Thus , w e ma y assume that dim X ≥ 2. W e reduce the pro of to the case where dim X = 2. Let V b e an aﬃne subspace of X of maximal dimension con taining L and so that V ∩ A = ∅ . Pic k an origin O ∈ L in X , and consider the 3.1. SEP ARA TION THEOREMS AND F ARKAS LEMMA 27 A L H Figure 3.2: Hahn-Banach Theorem, geometric form (Theorem 3.6) v ector space X O . W e would lik e to pro v e that V is a hyperplane, i.e., dim V = dim X − 1. W e pro ceed by con tradiction. Th us, assume that dim V ≤ dim X − 2. In this case, the quotien t space X/V has dimens ion at least 2. W e also know that X/V is isomorphic to the orthogonal compleme nt, V ⊥ , of V so w e ma y iden tify X/V and V ⊥ . The (orthogonal) pro jection map, π : X → V ⊥ , is linear, contin uous, and w e can sho w that π maps the op en subset A to an op en subse t π ( A ), whic h is also conv ex (one wa y to prov e that π ( A ) is op en is to observ e that for an y p oin t, a ∈ A , a small op en ball of cen ter a con tained in A is pro jected b y π to an op en ball containe d in π ( A ) a nd as π is surjectiv e, π ( A ) is op en). F urthermore, 0 / ∈ π ( A ). Since V ⊥ has dimens ion at least 2, there is some plane P (a subs pace o f dimension 2) inters ecting π ( A ), and th us, w e o btain a nonempty op en and con v ex subset B = π ( A ) ∩ P in the plane P ∼ = A 2 . So, we can apply Lemma 3.5 to B and the p oin t O = 0 in P ∼ = A 2 to ﬁnd a line, l , (in P ) through O with l ∩ B = ∅ . But then, l ∩ π ( A ) = ∅ and W = π − 1 ( l ) is an aﬃne subsp ace suc h that W ∩ A = ∅ and W prop erly contains V , contradicting the maximalit y of V . Remark: The geometric f o rm of t he Hahn-Banac h theorem also holds when the dimension of X is inﬁnite but a sligh tly more sophisticated pro of is requi red. Actually , a ll that is neede d is to prov e that a maximal aﬃne subspace con taining L and disjoin t from A exists. This can b e done using Zorn’s lemma. F or ot her pro ofs, see Bourbaki [9], Chapter 2, V alen tine [41], Chapter 2, Barvinok [3], Chapter 2, or Lax [26], Chapter 3.  Theorem 3 .6 is false if w e omit the assum ption that A is op en. F or a counter-ex ample, let A ⊆ A 2 b e the union of the half space y < 0 with the closed segm ent [0 , 1] on the x -axis and let L b e the p oin t (2 , 0) on the b oundary of A . It is also false if A is closed! (Find a coun ter-example). Theorem 3.6 has man y imp ortant corollaries. F or exampl e, w e will ev en tually prov e that for a n y t w o nonempt y disjoin t conv ex sets, A and B , there is a h yp erplane separating A a nd 28 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES A L H Figure 3.3: Hahn-Banac h Theorem, second v ersion (Theorem 3.7) B , but this will tak e some work (recall the deﬁnition of a separating h yperplane giv en in Deﬁnition 2.1). W e b egin with the fo llowing v ersion of the Hahn-Banac h theorem: Theorem 3.7 (Hahn-Banach , se c ond version) L et X b e a (ﬁnite-dim ensional) aﬃne s p ac e, A b e a nonempty c onvex subset of X with nonempty in terior and L b e an a ﬃne subsp ac e of X so that A ∩ L = ∅ . Then, ther e is some hyp erplane , H , c ontaining L and sep ar ating L and A . Pr o of . Since A is con v ex, by Corollary 3 .2 , ◦ A is also con v ex. By hypothesis, ◦ A is nonempt y . So, w e can apply Theorem 3.6 to the nonempt y op en and con v ex ◦ A and to the aﬃne subsp ace L . W e get a hyperplane H containing L suc h t hat ◦ A ∩ H = ∅ . How ev er, A ⊆ A = ◦ A and ◦ A is con tained in the closed half space ( H + or H − ) con taining ◦ A , so H separates A and L . Corollary 3.8 Given an aﬃne sp ac e, X , let A and B b e two non e mpty di s joint c onvex subsets and assume that A has nonem pty interior ( ◦ A 6 = ∅ ). Then, ther e is a hyp erplane sep ar ating A and B . Pr o of . Pic k some origin O and consider the ve ctor space X O . Deﬁne C = A − B (a special case of the Mink o wski sum) a s follow s: A − B = { a − b | a ∈ A, b ∈ B } = [ b ∈ B ( A − b ) . It is easily v eriﬁed that C = A − B is con v ex and has nonempt y in terior (as a union of subsets ha ving a no nempty in terior). F urthermore O / ∈ C , since A ∩ B = ∅ . 1 (Note that the deﬁn ition 1 Readers who prefer a pur ely aﬃne ar gumen t may deﬁne C = A − B as the aﬃne subset A − B = { O + a − b | a ∈ A, b ∈ B } . 3.1. SEP ARA TION THEOREMS AND F ARKAS LEMMA 29 A B H Figure 3.4: Separation Theorem, v ersion 1 (Corollary 3 .8 ) dep ends o n t he choic e of O , but this has no eﬀect on the pro of.) Since ◦ C is nonempt y , w e can apply The orem 3.7 to C and to the aﬃne subspac e { O } and w e get a h yp erplane, H , separating C and { O } . Let f b e any linear form deﬁning the h yp erplane H . W e ma y assume that f ( a − b ) ≤ 0, for all a ∈ A and all b ∈ B , i.e. , f ( a ) ≤ f ( b ). Consequen tly , if w e let α = sup { f ( a ) | a ∈ A } (whic h mak es sense, since the set { f ( a ) | a ∈ A } is b ounded), w e ha v e f ( a ) ≤ α for all a ∈ A and f ( b ) ≥ α for all b ∈ B , whic h sho ws that the aﬃne hyperplane deﬁned b y f − α separates A and B . Remark: Theorem 3 .7 and Corollary 3.8 a lso hold in the inﬁnite dimensional case, see Lax [26], Chapter 3, or Barvinok, Chapter 3. Since a hyperplane, H , separating A and B a s in Corollary 3.8 is the b oundary of eac h of the t w o half–spaces that it determines, we also obtain t he following coro llary: Corollary 3.9 Given an aﬃne sp ac e, X , let A and B b e two nonempty disjoint op en and c on v ex subsets. Then, ther e is a hyp erplane strictly sep ar ating A and B .  Bew are that Corollary 3.9 fai l s for close d con v ex sets. How ev er, Corollary 3.9 holds if w e also assume that A (or B ) is compact. W e need to review the notion of distance from a p oin t to a subset. Let X b e a metric space with distance function, d . Giv en an y p oint, a ∈ X , and any nonempt y subs et, B , of X , w e let d ( a, B ) = inf b ∈ B d ( a, b ) Again, O / ∈ C and C is conv ex. By adjusting O we can pick the aﬃne form, f , deﬁning a separating h yp erplane, H , of C a nd { O } , so that f ( O + a − b ) ≤ f ( O ), for a ll a ∈ A a nd all b ∈ B , i.e. , f ( a ) ≤ f ( b ). 30 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES (where inf is the notation for least upp er bo und). No w, if X is an aﬃne space of dimension d , it can b e giv en a metric structure by giving the corresp onding v ector space a metric structure, for instance, the metric induced by a Euclidean structure. W e hav e the fo llowing imp ortant prop ert y: F or an y nonempt y closed subset, S ⊆ X (not neces sarily con ve x), and an y po int, a ∈ X , there is some point s ∈ S “ac hieving the distance from a to S ,” i.e., so that d ( a, S ) = d ( a, s ) . The pro of uses the fact that the distance function is con tin uous and that a con tin uous function attains its minim um on a compact set, and is left as an exercis e. Corollary 3.10 Given an aﬃne sp ac e, X , let A and B b e two nonempty disjoint close d and c on v ex subsets, with A c omp act. Then, ther e is a hyp erplane strictly sep ar ating A and B . Pr o of s k etch . First, w e pic k an origin O and we giv e X O ∼ = A n a Euclidean structure. Let d denote the associated distance. Giv en an y subsets A o f X , let A + B ( O , ǫ ) = { x ∈ X | d ( x, A ) < ǫ } , where B ( a, ǫ ) denotes the op en ball, B ( a, ǫ ) = { x ∈ X | d ( a, x ) < ǫ } , of cente r a and radius ǫ > 0. Note that A + B ( O , ǫ ) = [ a ∈ A B ( a, ǫ ) , whic h shows that A + B ( O , ǫ ) is op en; furthermore it is easy to see that if A is con v ex, then A + B ( O , ǫ ) is a lso con ve x. No w, the function a 7→ d ( a, B ) (where a ∈ A ) is con tin uous and since A is compact, it ac hiev es its minim um, d ( A, B ) = min a ∈ A d ( a, B ), at some p oin t, a , o f A . Sa y , d ( A, B ) = δ . Since B is closed, there is some b ∈ B so that d ( A, B ) = d ( a, B ) = d ( a, b ) and since A ∩ B = ∅ , w e m ust ha v e δ > 0 . Th us, if w e pic k ǫ < δ / 2, w e see that ( A + B ( O , ǫ )) ∩ ( B + B ( O , ǫ )) = ∅ . No w, A + B ( O , ǫ ) and B + B ( O , ǫ ) are op en, con v ex and disjoin t and w e conclude b y applyin g Corollary 3.9. A “ cute ” application of Corollary 3.10 is one of the man y v ersions of “F a r k as Lemma” (1893-189 4 , 1902), a basic result in the theory of linear programming. F or any v ector, x = ( x 1 , . . . , x n ) ∈ R n , and an y real, α ∈ R , write x ≥ α iﬀ x i ≥ α , for i = 1 , . . . , n . Lemma 3.11 (F arkas L emma, V ersion I) Given any d × n r e a l matrix, A , and any ve ctor, z ∈ R d , exactly one of the fol lowing alternatives o c curs: (a) The l i n e ar s ystem, Ax = z , has a sol ution, x = ( x 1 , . . . , x n ) , such that x ≥ 0 and x 1 + · · · + x n = 1 , or 3.1. SEP ARA TION THEOREMS AND F ARKAS LEMMA 31 (b) Ther e is some c ∈ R d and s o me α ∈ R such that c ⊤ z < α and c ⊤ A ≥ α . Pr o of . Let A 1 , . . . , A n ∈ R d b e the n p oin ts corresp onding to the columns of A . Then, either z ∈ conv( { A 1 , . . . , A n } ) or z / ∈ con v( { A 1 , . . . , A n } ). In the ﬁrst case, w e ha ve a conv ex com bination z = x 1 A 1 + · · · + x n A n where x i ≥ 0 and x 1 + · · · + x n = 1, so x = ( x 1 , . . . , x n ) is a solution satisfying (a). In the second case, b y Corollar y 3.10, there is a h yp erplane, H , strictly separating { z } and con v( { A 1 , . . . , A n } ), whic h is ob viously closed. In fact, observ e that z / ∈ con v ( { A 1 , . . . , A n } ) iﬀ there is a h yp erplane, H , suc h that z ∈ ◦ H − and A i ∈ H + , for i = 1 , . . . , n . As the aﬃne h yp erplane, H , is the zero lo cus of an equation of the form c 1 y 1 + · · · + c d y d = α, either c ⊤ z < α and c ⊤ A i ≥ α for i = 1 , . . . , n , that is, c ⊤ A ≥ α , or c ⊤ z > α and c ⊤ A ≤ α . In the sec ond case, ( − c ) ⊤ z < − α and ( − c ) ⊤ A ≥ − α , so (b) is satisﬁed b y either c and α or b y − c and − α . Remark: If we relax the requiremen ts on solutions of Ax = z and only require x ≥ 0 ( x 1 + · · · + x n = 1 is no longer required) then, in condition (b), w e can tak e α = 0. This is another vers ion of F ark as Lemma. In this case, instead of considerin g the conv ex hull of { A 1 , . . . , A n } w e are considering the con v ex cone, cone( A 1 , . . . , A n ) = { λA 1 + · · · + λ n A n | λ i ≥ 0 , 1 ≤ i ≤ n } , that is, w e are dropping the condition λ 1 + · · · + λ n = 1. F or this v ersion of F a r k as Lemma w e need the follow ing separation lemma: Prop osition 3.12 L et C ⊆ E d b e any close d c onvex c one w ith ve rtex O . T hen, for every p oi n t, a , not in C , ther e is a hyp erplane, H , p assing thr o ugh O sep ar ating a and C w i th a / ∈ H . Pr o of . Since C is closed and con v ex and { a } is compact and conv ex, b y Corollary 3.10, there is a h yp erplane, H ′ , strictly separating a and C . Let H b e the hyperplane through O parallel to H ′ . Since C and a lie in the t w o disjoint op en half-spaces determine d b y H ′ , the p oin t a cannot b elong to H . Suppose that some p o in t, b ∈ C , lies in the op en half-space determined b y H and a . Then, the line, L , through O and b inte rsects H ′ in some p oin t, c , and as C is a cone, the half line determined b y O and b is con tained in C . So, c ∈ C w ould b elong to H ′ , a contradic tion. Therefore, C is contained in the closed half-space determined b y H that do es not con tain a , as claimed. Lemma 3.13 (F arkas L emma, V ersion I I) Given any d × n r e al matrix, A , and any ve ctor, z ∈ R d , exactly one of the fol lowing alternatives o c curs: 32 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES (a) The line a r system, Ax = z , ha s a solution, x , such that x ≥ 0 , or (b) Ther e is some c ∈ R d such that c ⊤ z < 0 and c ⊤ A ≥ 0 . Pr o of . The pro of is analogous to the pro of of Lemma 3.11 except that it uses Prop osition 3.12 instead of Corollary 3.10 and either z ∈ cone( A 1 , . . . , A n ) or z / ∈ cone( A 1 , . . . , A n ). One can sho w that F ark as I I implies F a rk as I. Here is another v ersion of F ark as Lemma ha ving to do with a system of inequalitie s, Ax ≤ z . Although, this v ersion ma y seem w eak er that F ark as II, it is actually equiv a len t to it! Lemma 3.14 (F arkas L emma, V ersio n III ) Given any d × n r e al matrix, A , and any ve ctor, z ∈ R d , exactly one of the fol lowing alternatives o c curs: (a) The system of ine qualities, Ax ≤ z , has a solution, x , or (b) Ther e is some c ∈ R d such that c ≥ 0 , c ⊤ z < 0 and c ⊤ A = 0 . Pr o of . W e use t wo tricks from linear programming: 1. W e con v ert the system of inequalities, Ax ≤ z , in to a system of equ ations by in tro- ducing a v ector of “slack v aria bles”, γ = ( γ 1 , . . . , γ d ), where the system o f equations is ( A, I )  x γ  = z , with γ ≥ 0. 2. W e replace eac h “unconstrained v a riable”, x i , b y x i = X i − Y i , with X i , Y i ≥ 0. Then, the original system Ax ≤ z has a solution, x (unconstrained), iﬀ the system of equations ( A, − A, I )   X Y γ   = z has a solution with X , Y , γ ≥ 0. By F ark as I I, this sys tem has no solution iﬀ there ex ists some c ∈ R d with c ⊤ z < 0 and c ⊤ ( A, − A, I ) ≥ 0 , that is, c ⊤ A ≥ 0, − c ⊤ A ≥ 0, and c ≥ 0. Ho w ev er, these four conditions reduce to c ⊤ z < 0, c ⊤ A = 0 and c ≥ 0. Finally , w e ha v e the separation theorem announced earlier for arbitrary nonempt y conv ex subsets . Theorem 3.15 (Sep ar ation of disj o int c on vex sets) Given an aﬃne sp a c e, X , let A and B b e two nonempty disjoint c onvex subsets. Then, ther e is a hyp erplane sep ar ating A and B . 3.1. SEP ARA TION THEOREMS AND F ARKAS LEMMA 33 x − x A B A + x C A − x D H O Figure 3.5: Separation Theorem, ﬁnal v ersion (Theorem 3.15) Pr o of . The pro of is by descending induction on n = dim A . If dim A = dim X , w e kno w from Prop osition 3.3 that A has nonempt y in terior and w e conclude using Corollary 3.8. Next, asssume that the induction hy p othesis holds if dim A ≥ n and assume dim A = n − 1. Pic k an o rigin O ∈ A and let H b e a hy p erplane con taining A . Pic k x ∈ X outside H and deﬁne C = con v( A ∪ { A + x } ) where A + x = { a + x | a ∈ A } and D = con v ( A ∪ { A − x } ) where A − x = { a − x | a ∈ A } . Note tha t C ∪ D is con v ex. If B ∩ C 6 = ∅ and B ∩ D 6 = ∅ , then the conv exit y of B and C ∪ D implies that A ∩ B 6 = ∅ , a con tradiction. Without loss of generalit y , assume that B ∩ C = ∅ . Since x is outside H , we ha v e dim C = n and b y the induction h yp othesis, there is a hy p erplane, H 1 separating C and B . As A ⊆ C , w e see that H 1 also separates A and B . Remarks: (1) The reader should compare this pro of (from V alen tine [41], Chapter I I) with Berger’s pro of using compactne ss of the pro jectiv e space P d [6] (Corollary 11.4.7). (2) Ra ther than usin g the Hahn-Banac h theorem to deduce separation results, one may pro ceed diﬀeren tly and use the following in tuitiv ely obvious lemma, as in V alen tine [41] (Theorem 2.4): Lemma 3.16 If A and B ar e two no n empty c onvex sets such that A ∪ B = X and A ∩ B = ∅ , then V = A ∩ B is a h yp erplane. One can then deduce Corollaries 3.8 and Theorem 3.15. Y et another a pproa ch is follo w ed in Barvinok [3]. 34 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES (3) How can some of the a bov e results b e generalized to inﬁnite dimensional aﬃne spaces , especially Theorem 3.6 and Corollary 3.8? One approac h is to sim ultaneously relax the notio n of in terior and tigh ten a little the notion of clos ure, in a more “linear and less top ological” fashion, as in V alen tine [41]. Giv en an y subset A ⊆ X (where X may b e inﬁnite dimensional, but is a Hausdorﬀ top ological v ector space), say that a p oin t x ∈ X is l i n e arly ac c essi b le fr om A iﬀ there is some a ∈ A with a 6 = x and ] a, x [ ⊆ A . W e let lina A b e the set of all p oin ts linearly accessib le from A and lin A = A ∪ lina A . A p oin t a ∈ A is a c or e p oint of A iﬀ for eve ry y ∈ X , with y 6 = a , there is some z ∈ ] a, y [ , suc h that [ a, z ] ⊆ A . The set of all core p oin ts is denoted core A . It is not diﬃcult to prov e that lin A ⊆ A and ◦ A ⊆ core A . If A has nonempty in terior, then lin A = A and ◦ A = core A . Also, if A is conv ex, then core A and lin A are con v ex. Then, Lemma 3.16 still holds (where X is not necessarily ﬁnite dimensional) if w e redeﬁne V as V = lin A ∩ lin B and allow the p ossibilit y that V could b e X itself. Corollary 3.8 also holds in the general case if w e assume that core A is nonempt y . F or details, see V alen tine [41], Chapter I and I I. (4) Y et another approac h is to deﬁne the notion of an algebraically op en conv ex set, as in Barvinok [3]. A con v ex set, A , is algebr aic al ly op e n iﬀ the in tersection of A with ev ery line, L , is an op en in terv al, p ossibly empt y or inﬁnite at either end (or all of L ). An o p en con ve x set is algebraically op en. Then, the Hahn-Banac h theorem holds pro vided that A is an algebraically op en con v ex set and similarly , Corollary 3.8 also holds prov ided A is algebraically op en. F or details, see Barvinok [3], Chapter 2 and 3. W e do not know how the notion “a lg ebraically op en” relates to the concept of core. (5) Theorems 3.6, 3.7 and Corollary 3.8 are prov ed in Lax [26] using the notion of gauge function in the more general case where A has some core p oin t (but b ew are that Lax uses the terminology interior p oint instead of core p o in t!). An imp ortan t special case of separation is the case where A is con v ex and B = { a } , for some p oin t, a , in A . 3.2 Supp o rt ing Hyp erplanes and Mink o wski’s Prop o- sition Recall the deﬁnition of a supp orting h yp erplane given in D eﬁ nition 2.2. W e ha v e the follow ing imp ortan t prop osition ﬁrst prov ed b y Mink ow ski (1896): Prop osition 3.17 (Minkowski) L et A b e a nonempty close d and c onvex subset. Then, for every p oint a ∈ ∂ A , ther e is a supp orting hyp erpl a ne to A thr ough a . 3.3. POLARITY AND DUALITY 35 Pr o of . Let d = dim A . If d < dim X (i.e., A has empty in terior), t hen A is con tained in some aﬃne subspace V of dimension d < dim X , and an y h yp erplane con taining V is a supp orting h yp erplane f or ev ery a ∈ A . No w, assume d = dim X , so that ◦ A 6 = ∅ . If a ∈ ∂ A , then { a } ∩ ◦ A = ∅ . By Theorem 3.6, there is a h yp erplane H separating ◦ A and L = { a } . Ho w ev er, b y Corollary 3.2, since ◦ A 6 = ∅ and A is closed, w e hav e A = A = ◦ A. No w, the half–space con taining ◦ A is closed, a nd th us, it contains ◦ A = A . Therefore, H separates A and { a } .  Bew are that Prop osition 3.17 is false when the dimens ion of X is inﬁnite a nd when ◦ A = ∅ . The prop osition b elo w giv es a suﬃcien t condition for a closed subset to b e con v ex. Prop osition 3.18 L et A b e a close d subset with nonem pty interior. If ther e is a supp orting hyp erpla ne for every p oint a ∈ ∂ A , then A is c o n vex. Pr o of . W e leav e it as an exe rcise (see Berger [6], Proposition 11 .5.4).  The condition that A has nonempt y in terior is crucial! The propo sition b elo w c haracterizes closed conv ex sets in terms o f (closed) half–spaces. It is another in tuitiv e fa ct whose rigorous pro of is non trivial. Prop osition 3.19 L et A b e a nonempty close d and c onvex subset. Then, A is the interse c- tion of al l the clos e d half–sp a c es c ontaining it. Pr o of . Let A ′ b e the in tersec tion o f a ll the closed half–spaces con taining A . It is immediatel y c hec k ed that A ′ is closed and con v ex and that A ⊆ A ′ . Assume that A ′ 6 = A , and pic k a ∈ A ′ − A . The n, w e can apply Corollary 3.10 to { a } and A and w e ﬁnd a h yp erplane, H , strictly sep arating A and { a } ; this sho ws that A b elongs to one of the tw o half-spaces determined by H , y et a do es not b elong to the same half-space, con tradicting the deﬁnition of A ′ . 3.3 P olarit y and Dualit y Let E = E n b e a Euclidean space of dimensi on n . Pic k an y origin, O , in E n (w e ma y assume O = (0 , . . . , 0)). W e know t hat the inner pro duct on E = E n induces a dualit y b et w een E and its dual E ∗ (for example, see Chapter 6 , Section 2 of Ga llier [20]), namely , u 7→ ϕ u , where ϕ u is the linear form deﬁned by ϕ u ( v ) = u · v , for all v ∈ E . F o r geometric purposes, it is more con ve nien t to recast this duality as a correspondence b etw een p oin ts and h yp erplanes, using the notion of p olarit y with respect to the unit sphere , S n − 1 = { a ∈ E n | k Oa k = 1 } . 36 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES First, w e need the follow ing simple fact: F or ev ery h yp erplane, H , not passin g thro ugh O , there is a unique po int, h , so that H = { a ∈ E n | Oh · Oa = 1 } . Indeed, an y h yp erplane, H , in E n is the n ull set of some equation of the form α 1 x 1 + · · · + α n x n = β , and if O / ∈ H , then β 6 = 0. Th us, any hy p erplane, H , not passin g through O is deﬁned b y an equation of the form h 1 x 1 + · · · + h n x n = 1 , if w e set h i = α i /β . So, if w e let h = ( h 1 , . . . , h n ), w e see that H = { a ∈ E n | Oh · Oa = 1 } , as claimed. No w, a ss ume that H = { a ∈ E n | Oh 1 · Oa = 1 } = { a ∈ E n | Oh 2 · Oa = 1 } . The functions a 7→ Oh 1 · Oa − 1 and a 7→ Oh 2 · Oa − 1 are t w o aﬃne forms deﬁning the same h yp erplane , so there is a nonzero scalar, λ , so that Oh 1 · Oa − 1 = λ ( Oh 2 · Oa − 1) for all a ∈ E n (see Gallier [2 0], Chapter 2, Section 2.10). In particular, for a = O , w e ﬁnd that λ = 1, a nd so, Oh 1 · Oa = Oh 2 · Oa fo r all a, whic h implies h 1 = h 2 . This pro v es t he uniqueness of h . Using the ab o v e, w e mak e the follo wing deﬁnition: Deﬁnition 3.3 Giv en any p oin t, a 6 = O , the p olar hyp erplane of a (w.r.t. S n − 1 ) or d ual of a is the h yp erplane, a † , giv en b y a † = { b ∈ E n | Oa · Ob = 1 } . Giv en a h yp erplane, H , not con taining O , the p ole of H (w.r.t S n − 1 ) or dual of H is the (unique) p oin t, H † , so that H = { a ∈ E n | OH † · Oa = 1 } . 3.3. POLARITY AND DUALITY 37 a a † O b Figure 3.6: The p olar, a † , of a p oin t, a , outside the sphere S n − 1 W e often abbreviate p olar h yp erplane to p olar. W e immediately c hec k that a †† = a and H †† = H , so, w e obta in a bijectiv e correspondence betw een E n − { O } and the set of h yp erplanes not passing through O . When a is outside the sphere S n − 1 , there is a nice geometric interpetation for the p olar h yp erplane, H = a † . Indeed, in this case, since H = a † = { b ∈ E n | Oa · Ob = 1 } and k Oa k > 1, the h yp erplane H interse cts S n − 1 (along an ( n − 2)-dimensional sphere) and if b is an y p oint on H ∩ S n − 1 , w e claim that Ob and ba are orthogonal. This means that H ∩ S n − 1 is the set of p oin ts on S n − 1 where the lines through a and tangen t to S n − 1 touc h S n − 1 (they fo rm a cone tangen t to S n − 1 with ap ex a ). Indeed, a s Oa = Ob + ba and b ∈ H ∩ S n − 1 i.e., Oa · Ob = 1 and k Ob k 2 = 1, w e get 1 = Oa · Ob = ( Ob + ba ) · Ob = k Ob k 2 + ba · Ob = 1 + ba · Ob , whic h implies ba · Ob = 0. When a ∈ S n − 1 , the h yp erplane a † is tangen t to S n − 1 at a . Also, o bse rv e that for an y p oint, a 6 = O , and any h yp erplane, H , not passing through O , if a ∈ H , then, H † ∈ a † , i.e, the p ole, H † , of H belongs to the p olar, a † , of a . Indeed, H † is the unique p oin t so that H = { b ∈ E n | OH † · Ob = 1 } and a † = { b ∈ E n | Oa · Ob = 1 } ; since a ∈ H , w e hav e OH † · Oa = 1, whic h show s t hat H † ∈ a † . If a = ( a 1 , . . . , a n ), the equation of the p olar hyperplane, a † , is a 1 X 1 + · · · + a n X n = 1 . 38 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES Remark: As we noted, p olarit y in a Euclidean space suﬀers from the minor defect that the p olar of the origin is undeﬁne d and, similarly , the p ole of a h yp erplane through the origin do es not mak e sense. If w e embed E n in to the pro jectiv e space, P n , b y a dding a “h yp erplane at inﬁnity ” (a cop y of P n − 1 ), thereb y viewing P n as the disjoin t union P n = E n ∪ P n − 1 , then the p olarit y corresp ondenc e can b e deﬁned ev erywh ere. Indeed, the p olar o f the origin is the h yp erplane at inﬁnity ( P n − 1 ) and since P n − 1 can b e view ed a s t he set of hyperplanes through the origin in E n , the p ole of a h yp erplane through the origin is the corresp onding “p oin t at inﬁnit y” in P n − 1 . No w, w e w ould like to extend this corresp ondence to subsets of E n , in particular, to con v ex sets. G iv en a hy p erplane, H , not con ta ining O , we denote b y H − the closed half- space con taining O . Deﬁnition 3.4 Giv en any subset, A , of E n , the set A ∗ = { b ∈ E n | Oa · Ob ≤ 1 , for all a ∈ A } = \ a ∈ A a 6 = O ( a † ) − , is called the p olar dual or r e cipr o c al of A . F o r simplicit y o f notation, w e write a † − for ( a † ) − . Obse rve that { O } ∗ = E n , so it is con v enien t to set O † − = E n , ev en though O † is undeﬁned. By deﬁnition, A ∗ is con v ex ev en if A is not. F urthermore, note that (1) A ⊆ A ∗∗ . (2) If A ⊆ B , then B ∗ ⊆ A ∗ . (3) If A is conv ex and closed, t hen A ∗ = ( ∂ A ) ∗ . It follo ws immediately fro m (1) and (2) that A ∗∗∗ = A ∗ . Also, if B n ( r ) is the (closed) ball of radius r > 0 and cente r O , it is obv ious b y deﬁnition that B n ( r ) ∗ = B n (1 /r ). In Figure 3.7, the po la r dual of the p olygon ( v 1 , v 2 , v 3 , v 4 , v 5 ) is the p olygon show n in green. This p olygon is cut out b y the half- planes determ ined b y the p olars of the v ertices ( v 1 , v 2 , v 3 , v 4 , v 5 ) and con taining t he cen ter of the circle. These p olar lines are all easy to determine by dra wing for each v ertex, v i , the tangen t lines to the circle and joining the con tact p oin ts. The construction of the p olar of v 3 is sho wn in detail. Remark: W e c hose a diﬀeren t notation for po lar h yp erplanes and p olars ( a † and H † ) and p olar duals ( A ∗ ), to av oid the p oten tial confusion b et w een H † and H ∗ , where H is a h y- p erplane (or a † and { a } ∗ , where a is a p oin t). Inde ed, they are completely diﬀeren t! F or example, the pola r dual of a hy p erplane is either a line orthogonal to H through O , if O ∈ H , or a semi-inﬁnite line through O and orthogonal to H whose endp oin t is the p ole, H † , of H , 3.3. POLARITY AND DUALITY 39 v 1 v 2 v 3 v 4 v 5 Figure 3.7: The p olar dual of a p o lygon whereas, H † is a single point! Z iegler ([43], Chapter 2) use the notation A △ instead of A ∗ for the p olar dual of A . W e w ould like to in v estigate the dualit y induced b y the op eration A 7→ A ∗ . Unfortunately , it is not alw a ys the case that A ∗∗ = A , but this is true when A is close d and con v ex, as sho wn in the followin g proposition: Prop osition 3.20 L et A b e any subset of E n (with orig i n O ). (i) If A is b ounde d, then O ∈ ◦ A ∗ ; if O ∈ ◦ A , then A ∗ is b ounde d. (ii) If A i s a close d and c onvex subset c ontaini ng O , then A ∗∗ = A . Pr o of . (i) If A is b ounded, then A ⊆ B n ( r ) for some r > 0 large enough. Then, B n ( r ) ∗ = B n (1 /r ) ⊆ A ∗ , so that O ∈ ◦ A ∗ . If O ∈ ◦ A , then B n ( r ) ⊆ A for some r small enough, so A ∗ ⊆ B n ( r ) ∗ = B r (1 /r ) and A ∗ is b ounded. (ii) W e alw a ys hav e A ⊆ A ∗∗ . W e pro v e that if b / ∈ A , then b / ∈ A ∗∗ ; this sho ws that A ∗∗ ⊆ A and thus, A = A ∗∗ . Since A is closed and conv ex and { b } is compact (a nd con v ex!), b y Corollary 3.10, there is a h yperplane, H , strictly separating A and b and, in particular, O / ∈ H , a s O ∈ A . If h = H † is the p ole of H , w e hav e Oh · Ob > 1 and Oh · Oa < 1 , for all a ∈ A since H − = { a ∈ E n | Oh · Oa ≤ 1 } . This show s that b / ∈ A ∗∗ , since A ∗∗ = { c ∈ E n | Od · Oc ≤ 1 for all d ∈ A ∗ } = { c ∈ E n | ( ∀ d ∈ E n )(if Od · Oa ≤ 1 for all a ∈ A, then Od · Oc ≤ 1) } , 40 CHAPTER 3. SEP ARA TION AND SUPPOR TING HYPERPLANES just let c = b and d = h . Remark: F or an arbitrary subse t, A ⊆ E n , it can b e sho wn that A ∗∗ = con v ( A ∪ { O } ), the top ological closure of the con v ex h ull o f A ∪ { O } . Prop osition 3.20 will pla y a k ey role in studying p olytopes, but b efore doing this, w e need one more prop osition. Prop osition 3.21 L et A b e any close d c on vex subset of E n such that O ∈ ◦ A . The p olar hyp erpla nes of the p oints of the b oundary of A c onstitute the set of s upp orting hyp erplanes of A ∗ . F urthermor e , for any a ∈ ∂ A , the p oints of A ∗ wher e H = a † is a supp orting hyp erplane of A ∗ ar e the p o l e s of s upp orting hyp erpl a nes of A at a . Pr o of . Since O ∈ ◦ A , w e ha v e O / ∈ ∂ A , and so, for ev ery a ∈ ∂ A , the p olar hyperplane a † is w ell-deﬁned. Pic k an y a ∈ ∂ A and let H = a † b e its p olar hy p erplane. By deﬁnition, A ∗ ⊆ H − , the closed half-space determ ined b y H and con taining O . If T is an y supporting h yp erplane to A at a , as a ∈ T , w e ha v e t = T † ∈ a † = H . F urthermore, it is a simpl e exercise to prov e that t ∈ ( T − ) ∗ (in fact, ( T − ) ∗ is the in terv al with endp oin ts O and t ). Since A ⊆ T − (b ecause T is a supp orting hy p erplane to A at a ), w e deduce that t ∈ A ∗ , and th us, H is a suppor ting h yp erplane to A ∗ at t . By Prop osition 3.20, as A is closed and con v ex, A ∗∗ = A ; it follo ws that all supp orting h yp erplanes to A ∗ are indeed obtained this w a y . Chapter 4 P olyhedra and P oly top es 4.1 P olyhedra, H -P olytop es and V -P olytop e s There are t w o na tura l wa ys to deﬁne a con v ex p olyhedron, A : (1) As the conv ex h ull of a ﬁnite set of p oin ts. (2) As a subset of E n cut out b y a ﬁnite n um b er of h yp erplanes, more precisely , as the in tersection of a ﬁnite n um b er of (closed) half- space s. As stated, these tw o deﬁnitions are not equiv alen t b ecause (1) implies that a p olyhedron is b ounded, whereas (2) allows un b ounded subsets. Now , if w e require in (2) that the conv ex set A is b ounded, it is quite clear for n = 2 that the t w o deﬁnitions (1) and (2) are equiv alen t; for n = 3, it is intui tive ly clear that deﬁnitions (1) and (2) are still equiv alen t, but proving this equiv alence rigorously do es not app ear to b e that easy . What ab out the equiv alence when n ≥ 4? It turns out that deﬁnitions (1) and (2) are equiv alen t for all n , but this is a nontrivial theorem and a rigorous pro of do es not come by so c heaply . F ortunately , since we hav e Krein and Milman’s theorem at our disp osal and p olar dualit y , w e can giv e a rather short pro of. The hard direction of the equiv alence consists in proving that deﬁnition (1) implies deﬁnition (2). This is where the dualit y induced b y p olarit y b ecomes handy , esp ecially , the fact that A ∗∗ = A ! (under the righ t h yp othese s). First, w e giv e precise deﬁni tions (followin g Z iegler [43]). Deﬁnition 4.1 Let E b e an y aﬃne Euclidean space of ﬁnite dimension, n . 1 An H -p olyhe dr on in E , for short, a p olyhe d r on , is an y subset, P = T p i =1 C i , of E deﬁned as the inters ection of a ﬁnite n um b er of closed half-spaces, C i ; an H -p olytop e in E is a b ounded po lyhe dron and a V -p olytop e is the con v ex hull, P = conv( S ), of a ﬁnite set of p o in t s, S ⊆ E . 1 This mea ns that the v ector spa ce, − → E , asso ciated with E is a Euclidean space. 41 42 CHAPTER 4. POL YHEDRA AND POL YTOPES (a) (b) Figure 4.1: (a) An H -p olyhedron. (b) A V -p olytope Ob viously , p olyhedra and p olytop es are con v ex and closed (in E ). Since the notions of H -p olytop e and V - p olytop e are equiv alen t (see Theorem 4.7), w e often use the simpler lo cution p olytop e. Examples of a n H -p olyhedron and of a V -po lytope are show n in Figure 4.1. Note that Deﬁnition 4 .1 allows H - polytop es and V -p olytop es to hav e an empty inte rior, whic h is somewhat of an inconv enience. This is not a problem, since w e may alw ay s restrict ourselv es to the aﬃne hull of P (some aﬃne space, E , of dimension d ≤ n , where d = dim( P ), as in Deﬁnition 3.1) as w e no w show . Prop osition 4.1 L et A ⊆ E b e a V -p olytop e or an H -p olyhe dr on, le t E = aﬀ ( A ) b e the aﬃne hul l of A in E (with the Euclide an structur e on E induc e d by the Euclide an structur e on E ) and write d = dim( E ) . Then, the f o l lowing ass e rtions hold: (1) The set, A , is a V -p olytop e in E (i.e., viewe d as a subset of E ) iﬀ A is a V -p olytop e in E . (2) The set, A , is an H -p olyhe dr on in E (i.e., viewe d as a s ubs e t of E ) iﬀ A is an H - p ol yhe dr on in E . Pr o of . (1) This follo ws immediately b ecause E is an aﬃne subspace of E and eve ry aﬃne subspace of E is closed under aﬃne com binations a nd so, a fortiori , under con v ex com bina- tions. W e lea v e the details as an easy exercise. (2) Ass ume A is an H -po lyhe dron in E and that d < n . By deﬁnition, A = T p i =1 C i , where the C i are closed half-spaces dete rmined b y some hy p erplanes, H 1 , . . . , H p , in E . (Observ e that the h yperplanes, H i ’s, associated with the closed half-spaces, C i , ma y not b e distinct. 4.1. POL YHEDRA, H -POL YTOPES AND V -POL YTOPES 43 F o r example, w e ma y hav e C i = ( H i ) + and C j = ( H i ) − , for t he t w o closed half-spaces determined b y H i .) As A ⊆ E , w e ha v e A = A ∩ E = p \ i =1 ( C i ∩ E ) , where C i ∩ E is one of the closed half- spaces determined b y the h yp erplane, H ′ i = H i ∩ E , in E . Th us, A is also an H -p olyhedron in E . Con v ersely , assume that A is an H - polyhedron in E and that d < n . As any h yp erplane, H , in E can b e written as the in tersection, H = H − ∩ H + , of the t w o closed half-spaces that it b ounds, E itself can b e written as the in tersection, E = p \ i =1 E i = p \ i =1 ( E i ) + ∩ ( E i ) − , of ﬁnitely man y half-spaces in E . No w, as A is an H - polyhedron in E , w e hav e A = q \ j =1 C j , where the C j are closed half- space s in E determine d b y some h yp erplanes, H j , in E . How ev er, eac h H j can b e extended to a h yp erplane, H ′ j , in E , and so, eac h C j can b e extended to a closed half-space, C ′ j , in E and we still ha v e A = q \ j =1 C ′ j . Conseque ntly , w e get A = A ∩ E = p \ i =1 (( E i ) + ∩ ( E i ) − ) ∩ q \ j =1 C ′ j , whic h pro ve s that A is also a n H -p olyhed ron in E . The follo wing simple prop osition sho ws that w e ma y assume that E = E n : Prop osition 4.2 Given any two aﬃn e Euclide an s p ac es, E and F , if h : E → F is any aﬃne m ap then: (1) If A i s any V -p olytop e in E , then h ( E ) is a V -p olytop e in F . (2) If h is bije ctive and A is any H -p olyhe dr on in E , then h ( E ) is an H -p olyhe dr o n in F . 44 CHAPTER 4. POL YHEDRA AND POL YTOPES Pr o of . (1) As an y aﬃne map preserv es aﬃne com binations it also preserv es con v ex com bi- nation. Th us, h (conv( S )) = conv( h ( S )), for an y S ⊆ E . (2) Sa y A = T p i =1 C i in E . Consider an y half-space, C , in E and assume that C = { x ∈ E | ϕ ( x ) ≤ 0 } , for some aﬃne form, ϕ , deﬁning the h yp erplane, H = { x ∈ E | ϕ ( x ) = 0 } . Then, as h is bijectiv e, w e get h ( C ) = { h ( x ) ∈ F | ϕ ( x ) ≤ 0 } = { y ∈ F | ϕ ( h − 1 ( y )) ≤ 0 } = { y ∈ F | ( ϕ ◦ h − 1 )( y ) ≤ 0 } . This shows that h ( C ) is one of the close d half-spaces in F determined by the h yperplane, H ′ = { y ∈ F | ( ϕ ◦ h − 1 )( y ) = 0 } . F urthermore, as h is bijectiv e, it preserv es in tersections so h ( A ) = h p \ i =1 C i ! = p \ i =1 h ( C i ) , a ﬁnite in tersection of closed half-spaces. Therefore, h ( A ) is an H -p olyhedron in F . By Prop osition 4.2 we ma y a ss ume that E = E d and b y Prop o sition 4.1 w e may assume that dim ( A ) = d . These prop ositions justify the t yp e of argumen t b eginning with: “W e ma y assume that A ⊆ E d has dimension d , that is, that A has nonempt y in terior”. This kind of reasonning will o ccur many times. Since the b oundary of a closed half-space, C i , is a h yp erplane, H i , and since h yp erplanes are deﬁned by aﬃne forms, a closed half-space is deﬁne d b y the lo cus of p oin ts satisfying a “linear” inequalit y of the form a i · x ≤ b i or a i · x ≥ b i , f or some v ector a i ∈ R n and some b i ∈ R . Since a i · x ≥ b i is equiv alen t to ( − a i ) · x ≤ − b i , w e may restrict our atten tion to inequalities with a ≤ sign. Th us, if A is the d × p matrix whose i th ro w is a i , w e see that the H -p olyhedron, P , is deﬁned b y the system of linear inequalities, Ax ≤ b , where b = ( b 1 , . . . , b p ) ∈ R p . W e write P = P ( A, b ) , w ith P ( A, b ) = { x ∈ R n | Ax ≤ b } . An equation, a i · x = b i , ma y b e handled as the conjunction of the t w o inequalities a i · x ≤ b i and ( − a i ) · x ≤ − b i . Also, if 0 ∈ P , observ e that w e m ust ha v e b i ≥ 0 for i = 1 , . . . , p . In this case, ev ery inequalit y for which b i > 0 can be normalized by dividing b oth sides b y b i , so w e ma y assume that b i = 1 or b i = 0. This observ ation will b e useful to sho w that the p olar dual of an H -po lyhe dron is a V -p olyhedron. Remark: Some authors call “con v ex” p olyhedra a nd “conv ex” p olytop es what we hav e simply called p o lyhe dra and p olytop es. Since Deﬁnition 4.1 implies that these ob jects a r e 4.1. POL YHEDRA, H -POL YTOPES AND V -POL YTOPES 45 Figure 4.2: Example of a p olytop e (a do decahed ron) con v ex and since w e are not going to consider non-con v ex p olyhedra in this chapte r, we stic k to the simpler terminology . One should consult Ziegler [43], Berger [6], Grun baum [24] and esp ecially Crom w ell [14], for pictures of p olyhedra and p olytop es. Figure 4.2 sho ws the picture a p olytope whose faces are all p en tagons. This p olytop e is called a do de c ahe dr on . The do decahedron has 12 fa ces, 30 edges and 20 v ertices. Ev en b etter and a lot more entertainin g, ta k e a lo ok at the sp ectac ular w eb sites of George Hart, Virtual Polye dr a : h ttp://www.georgehart.com/virtual-p olyhedra/vp .htm l, Ge or ge Hart ’s w eb site: h ttp://www.georgehart.com/ and also Zvi Har’El ’s w eb site: h ttp://www.math.tec hnion.ac.il/ rl/ The Uniform Polyhe dr a w eb site: http://www .mathconsult.c h/sho wro om/unip oly/ Pap er Mo dels of Polyhe dr a : h ttp://www.k ort halsaltes.com/ Bulato v’s Polyhe dr a Col l e ction : http://www .ph ysics.orst.edu/ bulatov /p olyhedra/ P aul Gett y’s Polyhe dr al Solids : h ttp://home.telep ort.com/ tpgett ys/p oly .sh tml Jill Britton’s Polyhe dr a Pa s times : h ttp://ccins.camosun.bc.ca/ jbritton/j bp olyhedra.h tm and man y other w eb sites dealing with polyhedra in one w ay or another by searc hing for “p olyhedra” on Go o g l e ! Ob viously , an n - simp lex is a V -p olytop e. The standar d n -cub e is the set { ( x 1 , . . . , x n ) ∈ E n | | x i | ≤ 1 , 1 ≤ i ≤ n } . 46 CHAPTER 4. POL YHEDRA AND POL YTOPES The standard cub e is a V -p olytop e. The standar d n -cr oss-p olytop e (or n -c o-cub e ) is the set { ( x 1 , . . . , x n ) ∈ E n | n X i =1 | x i | ≤ 1 } . It is also a V -p olytop e. What happ ens if we take the dual of a V -p olytop e (resp. a n H -p olytop e)? The follow ing prop osition, although very simple , is an imp o rtan t step in answ ering the ab o v e question: Prop osition 4.3 L et S = { a i } p i =1 b e a ﬁnite set of p oints in E n and l e t A = con v ( S ) b e its c on v ex hul l. If S 6 = { O } , then, the dual, A ∗ , of A w . r.t. the c e nter O is an H -p olyhe dr on; furthermor e, if O ∈ ◦ A , then A ∗ is an H -p olytop e, i.e., the dual o f a V -p olytop e with n o nempty interior is an H -p olytop e. If A = S = { O } , then A ∗ = E d . Pr o of . By deﬁnition, w e ha v e A ∗ = { b ∈ E n | Ob · ( p X j =1 λ j Oa j ) ≤ 1 , λ j ≥ 0 , p X j =1 λ j = 1 } , and the righ t hand side is clearly equal to T p i =1 { b ∈ E n | Ob · Oa i ≤ 1 } = T p i =1 ( a † i ) − , whic h is a polyhedron. (Recall that ( a † i ) − = E n if a i = O .) If O ∈ ◦ A , then A ∗ is b ounded (by Prop osition 3.20) and so, A ∗ is an H -p olytop e. Th us, the dual of the con ve x hu ll of a ﬁnite set o f p oin ts, { a 1 , . . . , a p } , is the in tersection of the half-spaces con taining O determined b y the p olar h yp erplanes of the p oin ts a i . It is con v enien t to r estate Proposition 4.3 using matrices. First, observ e t hat the pro of of Prop osition 4.3 shows that con v( { a 1 , . . . , a p } ) ∗ = con v ( { a 1 , . . . , a p } ∪ { O } ) ∗ . Therefore, w e ma y assume that not all a i = 0 (1 ≤ i ≤ p ). If w e pic k O as an origin, then ev ery p oin t a j can b e iden tiﬁed with a v ector in E n and O corresp o nds to the zero v ector, 0. Observ e that any set of p p oin ts, a j ∈ E n , corresponds to the n × p matrix, A , whose j th column is a j . Then, the equation of the the p olar h yp erplane, a † j , of an y a j ( 6 = 0) is a j · x = 1, that is a ⊤ j x = 1 . Conseque ntly , the system of inequalities deﬁning conv( { a 1 , . . . , a p } ) ∗ can b e written in matrix form as con v( { a 1 , . . . , a p } ) ∗ = { x ∈ R n | A ⊤ x ≤ 1 } , where 1 denotes the v ector of R p with all co ordinates equal to 1. W e write P ( A ⊤ , 1 ) = { x ∈ R n | A ⊤ x ≤ 1 } . There is a useful conv erse of this property as pro v ed in the next prop osition. 4.1. POL YHEDRA, H -POL YTOPES AND V -POL YTOPES 47 Prop osition 4.4 Given any s et o f p p oints, { a 1 , . . . , a p } , in R n with { a 1 , . . . , a p } 6 = { 0 } , if A is the n × p matrix whose j th c olumn is a j , then con v( { a 1 , . . . , a p } ) ∗ = P ( A ⊤ , 1 ) , with P ( A ⊤ , 1 ) = { x ∈ R n | A ⊤ x ≤ 1 } . Conversely, given any p × n matrix, A , not e qual to the zer o matrix, w e have P ( A, 1 ) ∗ = conv( { a 1 , . . . , a p } ∪ { 0 } ) , wher e a i ∈ R n is the i th r ow of A or, e q uiva lently, P ( A, 1 ) ∗ = { x ∈ R n | x = A ⊤ t, t ∈ R p , t ≥ 0 , I t = 1 } , wher e I is the r ow ve ctor of len gth p whose c o or dinates ar e al l e qual to 1 . Pr o of . Only the second part needs a pro of. Let B = con v ( { a 1 , . . . , a p } ∪ { 0 } ), where a i ∈ R n is the i th r ow of A . Then, b y the ﬁrst part, B ∗ = P ( A, 1 ) . As 0 ∈ B , b y Prop osition 3.20, w e hav e B = B ∗∗ = P ( A, 1 ) ∗ , as claimed. Remark: Prop osition 4.4 still holds if A is the zero matrix b ecause then, the inequalities A ⊤ x ≤ 1 (or Ax ≤ 1 ) are trivially satisﬁed. In the ﬁrst case, P ( A ⊤ , 1 ) = E d and in the second case, P ( A, 1 ) = E d . Using the ab o v e, the reader should c hec k that the dual of a simplex is a simplex and that the dual of an n -cube is an n -cross p olytop e. Observ e that not ev ery H -p olyhedron is of the form P ( A, 1 ). Firstly , 0 b elongs to the in terior of P ( A, 1 ) and, secondly cones with ap ex 0 can’t b e described in this form. How- ev er, w e will see in Section 4.3 that the full class of p olyhedra can be captured is we allo w inequalities of the form a ⊤ x ≤ 0. In order to ﬁnd the correspo nding “ V -deﬁnition” we will need to add p ositiv e combi nations of v ectors to conv ex com binations of p oin ts. In tuitiv ely , these v ectors corresp ond to “p oin ts at inﬁnit y”. W e will see shortly that if A is an H -p o lytope and if O ∈ ◦ A , then A ∗ is also a n H -p o lytope. F o r this, w e will pro v e ﬁrst that an H -p olytop e is a V -po lytope. This requires taking a closer lo ok a t po lyhe dra. Note that some of the h yp erplanes cutting out a p olyhed ron ma y b e redund ant. If A = T t i =1 C i is a p olyhedron (where each closed half-space, C i , is asso ciated with a h yper- plane, H i , so that ∂ C i = H i ), w e sa y that T t i =1 C i is an irr e dund a nt de c omp osition of A if A cannot b e expressed a s A = T m i =1 C ′ i with m < t (for some closed half- spaces, C ′ i ). The follo wing pro p osition sho ws that the C i in an irredundan t dec omp osition of A are unique ly determined b y A . 48 CHAPTER 4. POL YHEDRA AND POL YTOPES Prop osition 4.5 L et A b e a p olyhe dr on with nonempty interior and assume that A = T t i =1 C i is an irr e dundan t de c omp osition of A . Then, (i) Up to or der, the C i ’s ar e uniquely determine d by A . (ii) If H i = ∂ C i is the b ound a ry of C i , then H i ∩ A is a p olyhe dr on with nonempty interior in H i , denote d F acet i A , and c al le d a fac et of A . (iii) We have ∂ A = S t i =1 F a cet i A , wher e the union is i rr e dundant, i.e., F acet i A is n ot a subset of F acet j A , for al l i 6 = j . Pr o of . (ii) Fix any i and consid er A i = T j 6 = i C j . As A = T t i =1 C i is an irredundan t decompo- sition, there is some x ∈ A i − C i . Pic k any a ∈ ◦ A . By Lemma 3.1, w e get b = [ a, x ] ∩ H i ∈ ◦ A i , so b b elongs to t he interior of H i ∩ A i in H i . (iii) As ∂ A = A − ◦ A = A ∩ ( ◦ A ) c (where B c denotes the complem ent of a subset B of E n ) and ∂ C i = H i , w e get ∂ A = t \ i =1 C i ! − ◦ t \ j =1 C j ! = t \ i =1 C i ! − t \ j =1 ◦ C j ! = t \ i =1 C i ! ∩ t \ j =1 ◦ C j ! c = t \ i =1 C i ! ∩ t [ j =1 ( ◦ C j ) c ! = t [ j =1  t \ i =1 C i  ∩ ( ◦ C j ) c ! = t [ j =1 ∂ C j ∩  \ i 6 = j C i  ! = t [ j =1 ( H j ∩ A ) = t [ j =1 F a cet j A. If we had F acet i A ⊆ F acet j A , for some i 6 = j , then, by (ii), a s the aﬃne h ull of F acet i A is H i and the aﬃne h ull of F acet j A is H j , w e w ould hav e H i ⊆ H j , a con tradiction. (i) As the decomp o sition is irredundan t, the H i are pairwise distinct. Also, b y (ii), eac h facet, F acet i A , has dimension d − 1 (where d = dim A ). Then, in (iii), w e can sho w that the 4.1. POL YHEDRA, H -POL YTOPES AND V -POL YTOPES 49 decomposition of ∂ A as a union of p ot ytop es of dimension d − 1 whose pairwise nonempt y in tersections ha v e dimension at most d − 2 (since they are con tained in pairwise distinct h yp erplanes ) is unique up to p erm utation. Indeed, assume that ∂ A = F 1 ∪ · · · ∪ F m = G 1 ∪ · · · ∪ G n , where the F i ’s and G ′ j are p olyhe dra of dimension d − 1 and eac h of the unions is irredundan t. Then, w e claim that for each F i , there is some G ϕ ( i ) suc h that F i ⊆ G ϕ ( i ) . If not, F i w ould b e expres sed as a union F i = ( F i ∩ G i 1 ) ∪ · · · ∪ ( F i ∩ G i k ) where dim( F i ∩ G i j ) ≤ d − 2, since the h yp erplanes con taining F i and the G j ’s are pairwise distinct, whic h is absurd, since dim( F i ) = d − 1. By symmetry , for eac h G j , there is some F ψ ( j ) suc h that G j ⊆ F ψ ( j ) . But then, F i ⊆ F ψ ( ϕ ( i )) for all i and G j ⊆ G ϕ ( ψ ( j )) for all j whic h implies ψ ( ϕ ( i )) = i for all i and ϕ ( ψ ( j )) = j for all j since the unions are irredundan t. Th us, ϕ and ψ are m utual inv erses and the B j ’s are just a p erm utation of the A i ’s, a s claimed . Therefore, the facets, F acet i A , are uniquely determined b y A and so are the hy p erplanes, H i = aﬀ (F acet i A ), and the half-spaces, C i , that they determine. As a consequence , if A is a p olyhedron, then so are its f acets and the same holds for H -p olytop es. If A is an H -p olytope and H is a h yp erplane with H ∩ ◦ A 6 = ∅ , then H ∩ A is an H -p olytop e whose facets are of the form H ∩ F , where F is a facet of A . W e can use induction and deﬁne k -f a ces , for 0 ≤ k ≤ n − 1. Deﬁnition 4.2 Let A ⊆ E n b e a p olyhedron with nonempty in terior. W e deﬁne a k -fac e of A to b e a facet of a ( k + 1)-face of A , for k = 0 , . . . , n − 2, where an ( n − 1) -fac e is just a facet of A . The 1-faces are called e dges . Tw o k -faces are adjac ent if their inte rsection is a ( k − 1)-face. The p olyhedron A itself is also called a fac e (of itself ) or n -fac e and the k -faces of A with k ≤ n − 1 are called pr op er fac es of A . If A = T t i =1 C i is an irredundan t decomp osition o f A and H i is the b oundary of C i , then the hy p erplane, H i , is called the supp orting hyp erplane of the facet H i ∩ A . W e susp ect that the 0 -faces of a p olyhedron are v ertices in the sense of Deﬁnition 2.3. This is true and, in fact, the ve rtices of a p olyhedron coincide with its extreme p oin ts (see Deﬁnition 2.4). Prop osition 4.6 L et A ⊆ E n b e a p olyhe dr o n with nonempty interior. (1) F or an y p oint, a ∈ ∂ A , on the b oundary of A , the i nterse ction of al l the supp orting hyp erpla nes to A at a c oinc ides with the interse ction of al l the fac es that c ontain a . In p articular, p oints of or der k of A ar e those p oints in the r elative interior of the k -fac es of A 2 ; thus, 0 -fa c es c oincide with the vertic es of A . 2 Given a conv ex set, S , in A n , its re lative int erio r is its interior in the aﬃne h ull of S (which mig h t be of dimension strictly less than n ). 50 CHAPTER 4. POL YHEDRA AND POL YTOPES (2) The vertic es of A c oincide with the extr e me p oints of A . Pr o of . (1) If H is a supp orting hy p erplane to A at a , then, one of the half-spaces, C , determined by H , satisﬁes A = A ∩ C . It f o llo ws from Prop osition 4.5 that if H 6 = H i (where the h yp erplanes H i are the supp orting h yp erplanes of the facets of A ), then C is redundan t, from whic h (1) follows . (2) If a ∈ ∂ A is not extreme, then a ∈ [ y , z ], where y , z ∈ ∂ A . Ho w ev er, this implies that a has order k ≥ 1, i.e, a is not a vertex . 4.2 The Equiv alen ce of H -P olytop es and V -P olytop es W e are no w ready fo r the theorem show ing the equiv alence of V -p olytop es a nd H - p olytop es. This is a nontriv ial theorem usually attributed to W eyl and Mink owsk i (for example, see Barvinok [3]). Theorem 4.7 (Weyl-Minkowski) If A is an H -p olytop e, then A has a ﬁnite numb er of extr eme p oints (e qual to its vertic es) and A is the c on vex hul l of its set of vertic es; thus, an H -p olytop e is a V -p olytop e. Mor e over, A has a ﬁnite n umb er of k -fac es (for k = 0 , . . . , d − 2 , wher e d = dim( A ) ). Conversely, the c onvex hul l of a ﬁnite set of p oints is an H -p olytop e. As a c onse quenc e, a V -p ol ytop e is an H -p olytop e. Pr o of . By restricting ourselv es to the aﬃne h ull of A (some E d , with d ≤ n ) we may assume that A has nonempt y inte rior. Since a n H -p olytope has ﬁnitely many facets, w e deduce b y induction that an H -p olytop e has a ﬁnite n um b er of k -faces, for k = 0 , . . . , d − 2. In particular, a n H -p olytop e has ﬁnitely many v ertices. By prop osition 4.6, these v ertices are the extre me p oin ts of A and since an H -p olytop e is compact and con v ex, b y the theorem of Krein and Milman (Theorem 2.5), A is the con v ex h ull of its set of ve rtices. Con v ersely , again, w e may assume that A has nonempt y in terior b y restricting ourselv es to the aﬃne h ull of A . Then, pic k an origin, O , in the inte rior of A and consider the dual, A ∗ , of A . By Prop osition 4.3, the con v ex set A ∗ is a n H - polytop e. By the ﬁrst part of the pro of of Theorem 4.7, the H -p olytop e, A ∗ , is the conv ex h ull of its v ertices. Finally , as the h yp otheses of Prop osition 3.20 and Prop osition 4.3 (again) hold, we deduce that A = A ∗∗ is an H -p olytop e. In view of Theorem 4.7, we are justiﬁed in dropping the V or H in fron t of p olytop e, a nd will do so from no w o n. Theorem 4.7 has some in teresting corollaries regarding the dual of a p olytope. Corollary 4.8 If A is any p olytop e in E n such that the in terior of A c on tains the origin, O , then the dual, A ∗ , of A is also a p olytop e whose i n terior c ontains O and A ∗∗ = A . 4.3. THE EQUIV ALENCE OF H -POL YHEDRA AND V -POL YHEDRA 51 Corollary 4.9 If A is any p olytop e in E n whose interior c ontains the origin, O , then the k -fac e s of A ar e in bije ction with the ( n − k − 1) -fac es of the dual p olytop e, A ∗ . This c orr esp ondenc e is as fol lows: If Y = aﬀ ( F ) is the k -dimensional subsp ac e determining the k -fac e , F , of A then the subsp ac e, Y ∗ = aﬀ ( F ∗ ) , determin ing the c orr esp onding fac e, F ∗ , of A ∗ , is the interse c tion of the p olar hyp erplanes of p oints in Y . Pr o of . Immediate from Prop o sition 4.6 and Prop osition 3.21. W e also ha v e the follo wing prop osition whose pro of w ould not be that simple if w e only had the notion of a n H -p olytop e ( a s a matter of fact, there is a w a y of pro ving Theorem 4.7 using Prop osition 4.10) Prop osition 4.10 If A ⊆ E n is a p olytop e and f : E n → E m is an aﬃne map, then f ( A ) is a p olytop e in E m . Pr o of . Immediate, since an H -p olytop e is a V - polytop e and since aﬃne maps send conv ex sets to con v ex sets. The reader should c hec k that the Mink ows ki sum of p olytop es is a p olytop e. W e w ere able to giv e a short pro of of Theorem 4.7 b ecaus e w e relied on a p ow erful theorem, namely , Krein and Milman. A draw bac k of this approac h is that it by passes the in teresting and imp ortan t problem of designing algorithms for ﬁnding the ve rtices of an H -p olyhedron from the sets of inequalities deﬁning it. A method fo r doing this is F ourier- Motzkin elimination, see Ziegler [43] (Chapter 1) a nd Section 4.3. This is also a sp ecial case of line a r pr o gr amming . It is also p ossible to generalize the notion of V -p olytop e to p olyhedra using the notion of cone and to generalize the equiv alence theorem to H -p olyhedra and V - polyhedra. 4.3 The Equiv alen ce of H -P olyhed ra and V -P olyhedra The equiv alence o f H -p olytop es and V -p olytop es can b e generalized to p olyhedral sets, i.e. , ﬁnite inters ections of closed half-spaces that are not nece ssarily b ounded. This equiv alence w as ﬁrst pro v ed by Motzkin in the early 1930’s. It can b e pro v ed in sev eral w ays , some in v olving cones. Deﬁnition 4.3 Let E b e an y aﬃne Euclidean space of ﬁnite dimension, n (with asso ciated v ector space, − → E ). A subset, C ⊆ − → E , is a c one if C is closed under linear combinations in v olving only nonnnegativ e scalars called p ositive c ombinations . Giv en a subset, V ⊆ − → E , the c on i c al hul l or p ositive hul l of V is the set cone( V ) = n X I λ i v i | { v i } i ∈ I ⊆ V , λ i ≥ 0 for all i ∈ I o . 52 CHAPTER 4. POL YHEDRA AND POL YTOPES A V -p olyhe dr on or p olyhe d r al set is a subs et, A ⊆ E , suc h that A = con v( Y ) + cone( V ) = { a + v | a ∈ con v ( Y ) , v ∈ cone ( V ) } , where V ⊆ − → E is a ﬁnite set of v ectors and Y ⊆ E is a ﬁnite set of p oin ts. A set, C ⊆ − → E , is a V -c one or p olyhe d r al c one if C is the p ositiv e h ull of a ﬁnite set of v ectors, that is, C = cone( { u 1 , . . . , u p } ) , for some v ectors, u 1 , . . . , u p ∈ − → E . An H -c one is an y subs et of − → E giv en b y a ﬁnite in tersection of closed half-spaces cut out b y h yp erplanes through 0. The po sitiv e hull, cone( V ), of V is also denoted p os( V ). Observ e that a V -cone can be view ed as a p olyhedral set for whic h Y = { O } , a single po int. Ho w ev er, if we tak e the p oin t O as the origin, w e ma y view a V -p olyhedron, A , for whic h Y = { O } , as a V -cone. W e will switc h back and forth b et w een these t w o views of cones as w e ﬁnd it conv enien t, this should not cause an y confusion. In this sec tion, w e fav or the view that V - cones are sp ecial kinds of V -p olyhe dra. As a consequen ce, a ( V or H )-cone alwa ys con tains 0, sometimes called an ap e x of the cone. A set of the form { a + tu | t ≥ 0 } , where a ∈ E is a p o in t and u ∈ − → E is a nonzero vec tor, is called a half-line or r ay . Then, w e see that a V -p olyhedron, A = con v( Y ) + cone( V ), is the conv ex h ull of the union of a ﬁnite set of p oints with a ﬁnite set of ray s. In t he case of a V -cone, all these rays meet in a common p oin t, a n ap ex of the cone. Prop ositions 4.1 and 4.2 generalize easily t o V -p olyhedra a nd cones . Prop osition 4.11 L et A ⊆ E b e a V -p olyhe dr on or an H -p olyhe dr on, let E = aﬀ ( A ) b e the aﬃne hul l of A in E (with the Euclide an structur e on E induc e d by the Euclide an structur e on E ) and write d = dim( E ) . Then, the f o l lowing ass e rtions hold: (1) The set, A , is a V -p olyhe dr on in E (i.e., v iewe d as a subset of E ) iﬀ A is a V -p olyhe dr on in E . (2) The set, A , is an H -p olyhe dr on in E (i.e., viewe d as a s ubs e t of E ) iﬀ A is an H - p ol yhe dr on in E . Pr o of . W e already pro v ed (2) in Prop osition 4.1. F or (1), observ e that the direction, − → E , of E is a linear subsp ace of − → E . Conseque ntly , E is closed under aﬃne com binations and − → E is closed under linear com binations and the result follo ws immediately . Prop osition 4.12 Given any two aﬃne Euclide an sp ac es, E an d F , if h : E → F is any aﬃne m ap then: 4.3. THE EQUIV ALENCE OF H -POL YHEDRA AND V -POL YHEDRA 53 (1) If A i s any V -p olyhe dr on in E , then h ( E ) is a V -p o l yhe dr on in F . (2) If g : − → E → − → F is any line ar map and if C is any V - c one in − → E , then g ( C ) is a V -c o n e in − → F . (3) If h is bije ctive and A is any H -p olyhe dr on in E , then h ( E ) is an H -p olyhe dr o n in F . Pr o of . W e already pro v ed (3) in Prop osition 4.2. F or (1), using the fact that h ( a + u ) = h ( a ) + − → h ( u ) for an y aﬃne map, h , where − → h is the linear map asso ciated with h , w e get h (con v( Y ) + cone( V )) = con v ( h ( Y )) + cone( − → h ( V )) . F o r (1), as g is linear, we get g (cone( V )) = cone( g ( V )) , establishing the prop osition. Prop ositions 4.1 1 and 4.12 allow us t o assume that E = E d and that our ( V or H )- p olyhedra, A ⊆ E d , ha v e nonempt y in terior ( i.e. dim( A ) = d ). The generalization o f Theorem 4.7 is that ev ery V -p olyhed ron, A , is an H -p olyhedron and con v ersely . At ﬁrst g la nce , it may seem that there is a small problem when A = E d . Indeed , Deﬁnition 4.3 allow s the p ossibilit y that cone ( V ) = E d for some ﬁnite subset, V ⊆ R d . This is b ecause it is p ossible to generate a basis of R d using ﬁnitely man y p ositiv e com binations. On the other hand the deﬁnition of an H -p olyhe dron, A , (Deﬁnition 4.1) assumes that A ⊆ E n is cut out b y at le ast one h yp erplane. So, A is alw a ys con tained in some half-space of E n and strictly speaking, E n is not an H -p olyhedron! The simplest w ay to circum v en t this diﬃcult y is to decree that E n itself is a po lyh edron, whic h w e do. Y et another solution is to assume that, unless stated otherwi se, ev ery ﬁnite set of v ec- tors, V , that we consider when deﬁning a p olyhedron has the prop ert y that there is some h yp erplane, H , through the origin so that all the vec tors in V l ie in only one of the t w o closed half-spaces determin ed b y H . But then, the p olar dual of a p olyhe dron can’t b e a single p oin t! Therefore, w e stic k to our decision that E n itself is a po lyh edron. T o pro v e the equiv alence of H - polyhedra and V - polyhedra, Ziegler proceeds as follows : First, he sho ws that the equiv a lence of V -p olyhedra and H - polyhedra reduces to the equiv a - lence of V - cones and H -cones using an “old trick ” o f pro jectiv e g eometry , namely , “homoge- nizing” [43] (Chapter 1). Then, he uses t w o dual v ersions of F ourier-Motzkin elimination to pass from V -cones to H - cones and conv ersely . Since the homogenization metho d is an imp ortan t tec hnique w e will desc rib e it in some detail. Ho w ev er, it turns out that the double dualization tec hnique used in the pro of of Theorem 4.7 can b e easily adapted to prov e that ev ery V -p olyhedron is an H -p olyhedron. Moreo v er, it can also b e used to pro v e that ev ery H - p olyhedron is a V -p olyhe dron! So, 54 CHAPTER 4. POL YHEDRA AND POL YTOPES w e will not describe the v ersion of F ourier-Motzkin elimination used to go from V -cones to H -cones. Ho w ev er, w e will presen t the F o urier-Motzk in elimination method used to go from H -cones to V -cones. Here is the generalization of Prop osition 4.3 to p olyhedral sets. In order to av o id confusion b et w een the origin of E d and the cen ter of p olar duality w e will denote the origin b y O and the cen ter of our p olar duality by Ω. Giv en any nonzero v ector, u ∈ R d , let u † − b e the closed half-space u † − = { x ∈ R d | x · u ≤ 0 } . In other w ords, u † − is the closed half-space b ounded by the hy p erplane t hro ugh Ω normal to u and on the “opp osite side ” of u . Prop osition 4.13 L et A = con v( Y ) + cone ( V ) ⊆ E d b e a V -p olyhe dr on with Y = { y 1 , . . . , y p } and V = { v 1 , . . . , v q } . Then, for a n y p oint, Ω , if A 6 = { Ω } , then the p olar dual, A ∗ , of A w.r.t. Ω is the H -p olyhe dr o n given by A ∗ = p \ i =1 ( y † i ) − ∩ q \ j =1 ( v † j ) − . F urthermor e, if A has nonempty interior and Ω b elongs to the interior of A , then A ∗ is b ounde d, that is, A ∗ is an H -p olytop e. If A = { Ω } , then A ∗ is the sp e cial p olyhe dr on, A ∗ = E d . Pr o of . By deﬁnition of A ∗ w.r.t. Ω, we hav e A ∗ = ( x ∈ E d      Ωx · Ω p X i = 1 λ i y i + q X j = 1 µ j v j ! ≤ 1 , λ i ≥ 0 , p X i =1 λ i = 1 , µ j ≥ 0 ) = ( x ∈ E d      p X i =1 λ i Ωx · Ωy i + q X j =1 µ j Ωx · v j ≤ 1 , λ i ≥ 0 , p X i =1 λ i = 1 , µ j ≥ 0 ) . When µ j = 0 for j = 1 , . . . , q , w e get p X i =1 λ i Ωx · Ωy i ≤ 1 , λ i ≥ 0 , p X i =1 λ i = 1 and w e c hec k that ( x ∈ E d      p X i =1 λ i Ωx · Ωy i ≤ 1 , λ i ≥ 0 , p X i =1 λ i = 1 ) = p \ i =1 { x ∈ E d | Ωx · Ωy i ≤ 1 } = p \ i =1 ( y † i ) − . 4.3. THE EQUIV ALENCE OF H -POL YHEDRA AND V -POL YHEDRA 55 The p oin ts in A ∗ m ust also satisfy the conditions q X j =1 µ j Ωx · v j ≤ 1 − α , µ j ≥ 0 , µ j > 0 for some j , 1 ≤ j ≤ q , with α ≤ 1 (here α = P p i =1 λ i Ωx · Ωy i ). In particular, for ev ery j ∈ { 1 , . . . , q } , if we set µ k = 0 for k ∈ { 1 , . . . , q } − { j } , w e should hav e µ j Ωx · v j ≤ 1 − α for all µ j > 0 , that is, Ωx · v j ≤ 1 − α µ j for all µ j > 0 , whic h is equiv alen t to Ωx · v j ≤ 0 . Conseque ntly , if x ∈ A ∗ , w e m ust also ha v e x ∈ q \ j =1 { x ∈ E d | Ωx · v j ≤ 0 } = q \ j =1 ( v † j ) − . Therefore, A ∗ ⊆ p \ i =1 ( y † i ) − ∩ q \ j =1 ( v † j ) − . Ho w ev er, the reve rse inclus ion is ob vious and thus, w e ha v e equalit y . If Ω b elongs to the in terior of A , we know from Prop osition 3.20 that A ∗ is b ounded. Therefore, A ∗ is indeed an H -p olytop e of the ab ov e form. It is fruitful to restate Prop osition 4.13 in terms of matrices (as w e did for Prop osition 4.3). First, observ e that (con v ( Y ) + cone( V )) ∗ = (conv( Y ∪ { Ω } ) + cone( V )) ∗ . If we pic k Ω as an origin then w e can represen t the p oin ts in Y a s v ectors. The old origin is still denoted O and Ω is now denoted 0. The zero v ector is denoted 0 . If A = con v( Y ) + cone( V ) 6 = { 0 } , let Y b e the d × p matrix whose i th column is y i and let V is the d × q matrix whose j th column is v j . Then Prop osition 4.13 sa ys that (con v( Y ) + cone( V )) ∗ = { x ∈ R d | Y ⊤ x ≤ 1 , V ⊤ x ≤ 0 } . W e write P ( Y ⊤ , 1 ; V ⊤ , 0 ) = { x ∈ R d | Y ⊤ x ≤ 1 , V ⊤ x ≤ 0 } . If A = conv( Y ) + cone( V ) = { 0 } , then b oth Y and V m ust b e zero matrices but then, the inequalities Y ⊤ x ≤ 1 and V ⊤ x ≤ 0 are trivially satisﬁed b y all x ∈ E d , so ev en in this case w e ha ve E d = (conv( Y ) + cone( V )) ∗ = P ( Y ⊤ , 1 ; V ⊤ , 0 ) . The con v erse of Prop osition 4.13 also holds as sho wn b elo w. 56 CHAPTER 4. POL YHEDRA AND POL YTOPES Prop osition 4.14 L et { y 1 , . . . , y p } b e any set o f p oints in E d and let { v 1 , . . . , v q } b e any set of no n zer o ve ctors in R d . If Y is the d × p matrix whose i th c olumn is y i and V is the d × q matrix whose j th c olumn is v j , then (con v( { y 1 , . . . , y p } ) ∪ cone( { v 1 , . . . , v q } )) ∗ = P ( Y ⊤ , 1 ; V ⊤ , 0 ) , with P ( Y ⊤ , 1 ; V ⊤ , 0 ) = { x ∈ R d | Y ⊤ x ≤ 1 , V ⊤ x ≤ 0 } . Conversely, given any p × d matrix, Y , and any q × d matrix, V , we hav e P ( Y , 1 ; V , 0 ) ∗ = conv( { y 1 , . . . , y p } ∪ { 0 } ) ∪ cone( { v 1 , . . . , v q } ) , wher e y i ∈ R n is the i th r ow of Y and v j ∈ R n is the j th r ow of V or, e quivale n tly, P ( Y , 1 ; V , 0 ) ∗ = { x ∈ R d | x = Y ⊤ u + V ⊤ t, u ∈ R p , t ∈ R q , u, t ≥ 0 , I u = 1 } , wher e I is the r ow ve ctor of len gth p whose c o or dinates ar e al l e qual to 1 . Pr o of . Only the second part needs a pro of. Let B = conv( { y 1 , . . . , y p } ∪ { 0 } ) ∪ cone( { v 1 , . . . , v q } ) , where y i ∈ R p is the i th r ow of Y a nd v j ∈ R q is the j th r ow of V . Then, by the ﬁrst part, B ∗ = P ( Y , 1 ; V , 0 ) . As 0 ∈ B , b y Prop osition 3.20, w e hav e B = B ∗∗ = P ( Y , 1 ; V , 0 ), as claimed. Prop osition 4.14 has the follo wing imp ortan t Corollary: Prop osition 4.15 The fol lowing assertions hold: (1) The p olar dual, A ∗ , of every H -p olyhe d r on, is a V -p olyhe dr on. (2) The p olar dual, A ∗ , of every V -p olyhe dr on, is an H - p olyhe dr on. Pr o of . (1) W e ma y assum e that 0 ∈ A , in whic h case, A is of the form A = P ( Y , 1 ; V , 0 ). By the second part of Propo sition 4.1 4 , A ∗ is a V - polyhedron. (2) This is the ﬁrst part of Prop osition 4.14. W e can no w use Prop osition 4.13, Prop osition 3.20 and Krein and Millman’s Theorem to pro v e that ev ery V -p olyhedron is an H -p olyhedron. Prop osition 4.16 Every V -p olyhe dr on, A , is an H -p ol yhe dr on. F urthermor e, if A 6 = E d , then A is of the fo rm A = P ( Y , 1 ) . 4.4. F OURIER-MOTZKIN ELIMINA TION AND CONES 57 Pr o of . Let A b e a V -p olyhedron of dimension d . Th us, A ⊆ E d has nonempt y in terior so w e can pic k some p o in t , Ω, in the in terior of A . If d = 0, then A = { 0 } = E 0 and w e are done. Otherwise, b y Prop osition 4.1 3, the p olar dual, A ∗ , of A w.r.t. Ω is an H -p olytop e. Then, as in the pro o f of Theorem 4.7, using Krein and Millman’s Theorem w e deduc e that A ∗ is a V -p olytop e. No w, as Ω b elongs to A , b y Prop osition 3.20 (ev en if A is not b ounded) we ha v e A = A ∗∗ and b y Prop osition 4 .3 (or Prop osition 4.13) w e conclude that A = A ∗∗ is an H -p olyhedron of the form A = P ( Y , 1 ). In terestingly , w e can no w prov e easily that ev ery H -p olyhedron is a V -p olyhedron. Prop osition 4.17 Every H -p olyhe dr on is a V -p olyhe dr on. Pr o of . Let A b e an H -p olyhedron of dimension d . By Prop osition 4.15, the p olar dual, A ∗ , of A is a V -p olyhedron. By Prop osition 4.16, A ∗ is an H - polyhedron and again, b y Prop osition 4.15, w e deduce that A ∗∗ = A is a V - polyhedron ( A = A ∗∗ b ecause 0 ∈ A ). Putting together Prop ositions 4 .16 and 4 .1 7 we obtain o ur main theorem: Theorem 4.18 (Equivalenc e of H -p olyhe dr a an d V -p olyhe dr a) Every H -p olyhe dr on is a V - p ol yhe dr on and c on v ersely. Ev en though w e pro v ed the main result of this section, it is instructiv e to consider a more computational pro of making use of cones and an elimination method kno wn as F ourier- Motzkin elimination . 4.4 F ourier-Mot zkin Elimination and the P olyhedron - Cone Corre sp ondence The problem with the con v erse of Propo sition 4.16 when A is un b ounded ( i.e. , not compact) is tha t Krein and Millman’s Theorem do es not apply . W e need to take in to accoun t “p oin ts at inﬁnit y” corresponding to certain v ectors. The tric k w e used in Propo sition 4.16 is that the p olar dual of a V - polyhedron with nonempt y in terior is an H -p olytop e . This reduction to p olytop es a llo w ed us to use K rein and Millman t o con v ert an H -p olytop e to a V -p olytope and then again w e to ok the p olar dual. Another trick is to switc h to cones b y “homogenizing”. Give n an y subse t, S ⊆ E d , w e can form the cone, C ( S ) ⊆ E d +1 , b y “placing” a cop y of S in the h yp erplane , H d +1 ⊆ E d +1 , of equation x d +1 = 1, and drawin g all the half- lines from the origin through an y p oin t of S . If S is giv en b y m p olynomial inequalities of the form P i ( x 1 , . . . , x d ) ≤ b i , 58 CHAPTER 4. POL YHEDRA AND POL YTOPES where P i ( x 1 , . . . , x d ) is a p olynomial of total degree n i , this amoun ts to forming the new homogeneous inequalities x n i d +1 P i  x 1 x d +1 , . . . , x d x d +1  − b i x n i d +1 ≤ 0 together with x d +1 ≥ 0. In particular, if the P i ’s are linear forms (whic h means that n i = 1), then our inequalities are of the form a i · x ≤ b i , where a i ∈ R d is some v ector and the homogenized inequalities are a i · x − b i x d +1 ≤ 0 . It will b e con v enien t to formalize the pro cess of putting a cop y of a subset, S ⊆ E d , in the h yp erplane, H d +1 ⊆ E d +1 , of equation x d +1 = 1, as follo ws: F or ev ery point, a ∈ E d , let b a =  a 1  ∈ E d +1 and let b S = { b a | a ∈ S } . Ob viously , the map S 7→ b S is a bijection b et w een the subs ets of E d and the subsets of H d +1 . W e will use this bijection to identify S and b S and use the simpler notation, S , unles s confusion arises. Let’s apply this to p olyhedra. Let P ⊆ E d b e an H -p olyhedron. Then, P is cut out b y m h yp erplanes , H i , and for eac h H i , there is a nonzero v ector, a i , and some b i ∈ R so that H i = { x ∈ E d | a i · x = b i } and P is give n by P = m \ i =1 { x ∈ E d | a i · x ≤ b i } . If A denotes the m × d matrix whose i -th ro w is a i and b is the v ector b = ( b 1 , . . . , b m ), then w e can write P = P ( A, b ) = { x ∈ E d | Ax ≤ b } . W e “homogenize” P ( A, b ) as follo ws: Let C ( P ) b e the subs et of E d +1 deﬁned b y C ( P ) =  x x d +1  ∈ R d +1 | Ax ≤ x d +1 b, x d +1 ≥ 0  =  x x d +1  ∈ R d +1 | Ax − x d +1 b ≤ 0 , − x d +1 ≤ 0  . 4.4. F OURIER-MOTZKIN ELIMINA TION AND CONES 59 Th us, w e see that C ( P ) is the H -cone give n by the sys tem of ineq ualities  A − b 0 − 1   x x d +1  ≤  0 0  and that b P = C ( P ) ∩ H d +1 . Con v ersely , if Q is an y H -cone in E d +1 (in fact, an y H - polyhedron), it is clear that P = Q ∩ H d +1 is an H -p olyhedron in H d +1 ≈ E d . Let us no w assume that P ⊆ E d is a V -p olyhedron, P = con v ( Y ) + cone( V ), where Y = { y 1 , . . . , y p } and V = { v 1 , . . . , v q } . Deﬁne b Y = { b y 1 , . . . , b y p } ⊆ E d +1 , and b V = { b v 1 , . . . , b v q } ⊆ E d +1 , b y b y i =  y i 1  , b v j =  v j 0  . W e ch ec k immediately that C ( P ) = cone( { b Y ∪ b V } ) is a V -cone in E d +1 suc h that b P = C ( P ) ∩ H d +1 , where H d +1 is the h yp erplane of equation x d +1 = 1. Con v ersely , if C = cone( W ) is a V - cone in E d +1 , with w id +1 ≥ 0 for ev ery w i ∈ W , w e pro v e next that P = C ∩ H d +1 is a V -p olyhedron. Prop osition 4.19 (Polyhe d r on–Cone Corr e s p ondenc e) We have the fol lowing c orr esp on- denc e b etwe en p o l yhe dr a in E d and c ones in E d +1 : (1) F or any H -p olyhe dr on, P ⊆ E d , if P = P ( A, b ) = { x ∈ E d | Ax ≤ b } , wher e A is an m × d -matrix and b ∈ R m , then C ( P ) given by  A − b 0 − 1   x x d +1  ≤  0 0  is an H -c one in E d +1 and b P = C ( P ) ∩ H d +1 , wher e H d +1 is the hyp erplane of e quation x d +1 = 1 . Conversely, if Q is any H -c one in E d +1 (in fact, any H -p olyhe dr on), then P = Q ∩ H d +1 is an H - p olyhe dr on in H d +1 ≈ E d . (2) L et P ⊆ E d b e any V -p olyhe dr on, whe r e P = con v ( Y ) + cone( V ) with Y = { y 1 , . . . , y p } and V = { v 1 , . . . , v q } . D eﬁne b Y = { b y 1 , . . . , b y p } ⊆ E d +1 , and b V = { b v 1 , . . . , b v q } ⊆ E d +1 , by b y i =  y i 1  , b v j =  v j 0  . 60 CHAPTER 4. POL YHEDRA AND POL YTOPES Then, C ( P ) = cone( { b Y ∪ b V } ) is a V -c one in E d +1 such that b P = C ( P ) ∩ H d +1 , Conversely, if C = cone( W ) is a V -c one in E d +1 , with w i d +1 ≥ 0 f o r every w i ∈ W , then P = C ∩ H d +1 is a V -p olyhe dr on in H d +1 ≈ E d . Pr o of . W e already prov ed ev erything exc ept the last part of the propo sition. Let b Y =  w i w i d +1     w i ∈ W, w i d +1 > 0  and b V = { w j ∈ W | w j d +1 = 0 } . W e claim that P = C ∩ H d +1 = conv( b Y ) + cone( b V ) , and th us, P is a V -p olyhedron. Recall that an y elemen t, z ∈ C , can b e written as z = s X k =1 µ k w k , µ k ≥ 0 . Th us, w e hav e z = s X k =1 µ k w k µ k ≥ 0 = X w i d +1 > 0 µ i w i + X w j d +1 =0 µ j w j µ i , µ j ≥ 0 = X w i d +1 > 0 w i d +1 µ i w i w i d +1 + X w j d +1 =0 µ j w j , µ i , µ j ≥ 0 = X w i d +1 > 0 λ i w i w i d +1 + X w j d +1 =0 µ j w j , λ i , µ j ≥ 0 . No w, z ∈ C ∩ H d +1 iﬀ z d +1 = 1 iﬀ P p i =1 λ i = 1 (where p is the n um b er of elem en ts in b Y ), since the ( d + 1) th co ordinate of w i w i d +1 is equal to 1, and the a bov e sho ws that P = C ∩ H d +1 = conv( b Y ) + cone( b V ) , as claimed. 4.4. F OURIER-MOTZKIN ELIMINA TION AND CONES 61 By Prop osition 4.19, if P is an H -p olyhedron, t hen C ( P ) is an H -cone. If w e can pro v e that ev ery H -cone is a V -cone, then again, Prop osition 4.19 shows that b P = C ( P ) ∩ H d +1 is a V - polyhedron and so, P is a V -p olyhedron. Therefore, in order to pro v e that ev ery H -p olyhedron is a V -p olyhedron it suﬃces to sho w that ev ery H -cone is a V -cone. By a similar argumen t, Prop osition 4 .19 show s that in order to pro v e that ev ery V - p olyhedron is an H -p olyhed ron it suﬃces to sho w that ev ery V -cone is an H - cone. W e will not pro v e this direction ag ain since w e already hav e it b y Prop osition 4.16. It remains to pro v e that ev ery H -cone is a V -cone. Let C ⊆ E d b e an H -cone. Then, C is cut out by m hyperplanes, H i , through 0. F or eac h H i , t here is a nonzero v ector, u i , so that H i = { x ∈ E d | u i · x = 0 } and C is giv en by C = m \ i =1 { x ∈ E d | u i · x ≤ 0 } . If A denotes the m × d matrix whose i -th ro w is u i , then w e can write C = P ( A, 0) = { x ∈ E d | Ax ≤ 0 } . Observ e that C = C 0 ( A ) ∩ H w , where C 0 ( A ) =  x w  ∈ R d + m | Ax ≤ w  is an H -cone in E d + m and H w =  x w  ∈ R d + m | w = 0  is an aﬃne subspace in E d + m . W e claim that C 0 ( A ) is a V - cone. This follo ws by observing that for ev ery  x w  satisfying Ax ≤ w , w e can write  x w  = d X i =1 | x i | (sign( x i ))  e i Ae i  + m X j =1 ( w j − ( Ax ) j )  0 e j  , and then C 0 ( A ) = cone  ±  e i Ae i  | 1 ≤ i ≤ d  ∪  0 e j  | 1 ≤ j ≤ m  . Since C = C 0 ( A ) ∩ H w is no w the inters ection of a V -cone with a n aﬃne subspace , to pro v e that C is a V -cone it is enough to pro v e that the in tersection of a V -cone with a h yp erplane is also a V -cone. F or this, w e use F o urier-Motzkin elim i n ation . It suﬃce s to pro v e the result for a hyperplane, H k , in E d + m of equation y k = 0 (1 ≤ k ≤ d + m ). 62 CHAPTER 4. POL YHEDRA AND POL YTOPES Prop osition 4.20 (F ourier-Motzkin Elimination) Say C = cone( Y ) ⊆ E d is a V -c one . Then, the interse ction C ∩ H k (wher e H k is the hyp erplan e of e quation y k = 0 ) is a V -c one, C ∩ H k = cone( Y /k ) , with Y /k = { y i | y ik = 0 } ∪ { y ik y j − y j k y i | y ik > 0 , y j k < 0 } , the set of ve ctors obtaine d fr om Y by “eliminating the k -th c o or dinate”. Her e, e ach y i is a ve c tor in R d . Pr o of . The only non trivial direction is to prov e tha t C ∩ H k ⊆ cone( Y /k ). F or this, consider an y v = P d i =1 t i y i ∈ C ∩ H k , with t i ≥ 0 and v k = 0. Suc h a v can b e written v = X i | y ik =0 t i y i + X i | y ik > 0 t i y i + X j | y j k < 0 t j y j and as v k = 0, w e hav e X i | y ik > 0 t i y ik + X j | y j k < 0 t j y j k = 0 . If t i y ik = 0 for i = 1 , . . . , d , w e are done. Otherwise, w e can write Λ = X i | y ik > 0 t i y ik = X j | y j k < 0 − t j y j k > 0 . Then, v = X i | y ik =0 t i y i + 1 Λ X i | y ik > 0   X j | y j k < 0 − t j y j k   t i y i + 1 Λ X j | y j k < 0   X i | y ik > 0 t i y ik   t j y j = X i | y ik =0 t i y i + X i | y ik > 0 j | y j k < 0 t i t j Λ ( y ik y j − y j k y i ) . Since the k th co ordinate of y ik y j − y j k y i is 0, the ab ov e show s that any v ∈ C ∩ H k can b e written as a p o sitiv e com bination of v ectors in Y /k . As discuss ed ab o v e, Prop osition 4.20 implies (again!) Corollary 4.21 Every H -p o lyhe dr on is a V -p o lyhe dr on. Another w a y o f proving that ev ery V - polyhedron is an H -p olyhedron is to use cones. This metho d is in teresting in its o wn righ t so we discuss it brieﬂy . Let P = con v ( Y ) + cone ( V ) ⊆ E d b e a V -p olyhedron. W e can view Y as a d × p matrix whose i th column is the i th v ector in Y and V as d × q matrix whose j th column is the j th v ector in V . Then, we can write P = { x ∈ R d | ( ∃ u ∈ R p )( ∃ t ∈ R d )( x = Y u + V t, u ≥ 0 , I u = 1 , t ≥ 0) } , 4.4. F OURIER-MOTZKIN ELIMINA TION AND CONES 63 where I is the ro w v ector I = (1 , . . . , 1 ) | {z } p . No w, observ e that P can b e in terpreted as the pro jection of the H - p olyhedron, e P ⊆ E d + p + q , giv en by e P = { ( x, u, t ) ∈ R d + p + q | x = Y u + V t, u ≥ 0 , I u = 1 , t ≥ 0 } on to R d . Conse quen tly , if w e can prov e that the pro jection of an H -p olyhedron is also an H -p olyhedron, then we will ha v e prov ed that ev ery V -p olyhedron is an H -p olyhedron. Here again, it is p ossible that P = E d , but that’s ﬁne since E d has b een declared to b e a n H -p olyhedron. In view of Prop osition 4.19 and the discus sion that follo w ed, it is enough to pro v e that the pro jection of any H -cone is an H -cone. This can b e done b y using a t yp e of F ourier-Motzkin elimination dual to the metho d used in Proposition 4.2 0 . W e state the result without pro of and refer the in terested reader to Ziegler [43], Section 1.2–1.3, for full details. Prop osition 4.22 If C = P ( A, 0) ⊆ E d is an H -c one, then the pr oje c tion , pro j k ( C ) , onto the hyp erplane, H k , of e quation y k = 0 is given by pro j k ( C ) = elim k ( C ) ∩ H k , with elim k ( C ) = { x ∈ R d | ( ∃ t ∈ R )( x + te k ∈ P ) } = { z − te k | z ∈ P , t ∈ R } = P ( A /k , 0) and wher e the r ows of A /k ar e given by A /k = { a i | a i k = 0 } ∪ { a i k a j − a j k a i | a i k > 0 , a j k < 0 } . It should b e noted that b oth F ourier-Motzkin elimination metho ds generate a quadratic n um b er of new v ectors o r inequalities at eac h step and th us they lead to a com binatorial explosion. Therefore, these methods b ecome intractable rather quic kly . The problem is that man y of the new v ectors or inequalities are redund ant. The reore, it is imp ortan t to ﬁnd wa ys of detecting redundancies and there are v arious metho ds for doing so. Again, the in terested reader should consult Ziegler [43], Chapter 1. There is ye t another w a y of pro ving that an H -p olyhedron is a V -p olyhedron without using F ourier-Motzkin elimination. As w e already observ ed, Krein and Millman’s theorem do es not apply if our p olyhedron is un b ounded. Actually , the full p o we r of Krein and Millman’s theorem is not neede d to sho w that an H - polytop e is a V -po lytope. The crucial p oin t is that if P is an H -p olytope with nonempt y in terior, then every line, ℓ , through an y p oin t, a , in the in terior of P in tersects P in a line segmen t. This is b ecause P is compact and ℓ is closed, so P ∩ ℓ is a compact subset of a line th us, a closed in terv al [ b, c ] with b < a < c , as a is in the inte rior of P . Therefore, we can use induction on the dimension of P to sho w that ev ery p oin t in P is a con v ex com bination of vertic es of the f acets of P . Now, if P is un b ounded and cut out b y at least t w o half-spaces (so, P is not a half-space), then w e claim that for eve ry p oin t, a , in the in terior of P , there is some line through a that in tersec ts t w o facets of P . This is b ecause if we pic k t he origin in the in terior of P , w e may assu me that P is g iven by an irredundan t interse ction, P = T t i =1 ( H i ) − , and for any p oint, a , in the 64 CHAPTER 4. POL YHEDRA AND POL YTOPES in terior of P , there is a line, ℓ , through P in gener al p osition w.r.t. P , whic h means that ℓ is not parallel to an y of the h yp erplanes H i and in tersects all of them in distinct p oin ts (see Deﬁnition 7.2). F ortunately , lines in general p osition alw a ys exist, as sho wn in Proposition 7.3. Using this fact, w e can prov e the follo wing result: Prop osition 4.23 L et P ⊆ E d b e an H - p olyhe dr on, P = T t i =1 ( H i ) − (an irr e dundant de- c om p osition), wi th nonem p ty interior. If t = 1 , that is, P = ( H 1 ) − is a h alf-sp ac e, then P = a + cone( u 1 , . . . , u d − 1 , − u 1 , . . . , − u d − 1 , u d ) , wher e a is a ny p oint in H 1 , the ve ctors u 1 , . . . , u d − 1 form a b asis of the dir e ction of H 1 and u d is no rm al to (the di r e c tion of ) H 1 . (When d = 1 , P is the half-line, P = { a + tu 1 | t ≥ 0 } .) If t ≥ 2 , then ev e ry p oint, a ∈ P , c an b e written as a c onvex c o m bination, a = (1 − α ) b + αc ( 0 ≤ α ≤ 1 ), wher e b an d c b elong to two dis tinct fac ets, F an d G , of P and wher e F = con v ( Y F ) + cone( V F ) and G = conv( Y G ) + cone( V G ) , for some ﬁ n ite sets of p oints, Y F and Y G and some ﬁnite sets of v e ctors, V F and V G . (Note: α = 0 or α = 1 is al lowe d.) C o nse quently, e v ery H -p olyhe dr on is a V -p olyhe dr on. Pr o of . W e pro ceed by induction on the dimension, d , of P . If d = 1, then P is either a closed in terv al, [ b, c ], or a half-line, { a + tu | t ≥ 0 } , where u 6 = 0. In either case, t he prop osition is clear. F o r the induction step, assume d > 1. Since ev ery fa cet, F , of P has dimens ion d − 1, the induction h ypo t hesis holds fo r F , that is, there exist a ﬁnite set of p oin ts, Y F , and a ﬁnite set of v ectors, V F , so that F = con v ( Y F ) + cone( V F ) . Th us, ev ery p oin t o n the b oundary of P is of the desired fo r m. Next, pic k an y p oin t, a , in the in terior o f P . Then, f r o m o ur previous discu ssion, there is a line, ℓ , through a in general p osition w.r.t. P . Consequen tly , the in tersection p oin ts, z i = ℓ ∩ H i , of the line ℓ with the h yp erplanes supp orting the facets of P exist and are all distinct. If w e giv e ℓ an o rien tat io n, the z i ’s can b e sorted and there is a unique k suc h that a lies betw een b = z k and c = z k +1 . But then, b ∈ F k = F a nd c ∈ F k +1 = G , where F and G the facets o f P supp orted b y H k and H k +1 , and a = (1 − α ) b + α c , a con v ex com bination. Conseque ntly , ev ery p oin t in P is a mixed con v ex + p ositiv e com bination of ﬁnitely man y p oin ts asso ciated with the facets of P and ﬁnitely many ve ctors asso ciated with the directions of the supp orting h yp erplane s of the facets P . Con v ersely , it is easy to see that an y suc h mixed com bination mus t b elong to P and therefore, P is a V -p olyhedron. W e conclude this sec tion with a v ersion of F ark as Lemma for p olyhedral sets. Lemma 4.24 (F arkas L emma, V ersion IV) L et Y b e any d × p matrix a nd V b e an y d × q matrix. F or every z ∈ R d , exactly on e of the fol lowing alternatives o c curs: 4.4. F OURIER-MOTZKIN ELIMINA TION AND CONES 65 (a) Ther e ex i s t u ∈ R p and t ∈ R q , with u ≥ 0 , t ≥ 0 , I u = 1 and z = Y u + V t . (b) Ther e is some ve c tor, ( α, c ) ∈ R d +1 , such that c ⊤ y i ≥ α for al l i with 1 ≤ i ≤ p , c ⊤ v j ≥ 0 for al l j with 1 ≤ j ≤ q , and c ⊤ z < α . Pr o of . W e use F a rk as Lemma, V ersion I I (Lemma 3.13). Observ e that (a) is equiv alen t to the problem: Find ( u, t ) ∈ R p + q , so that  u t  ≥  0 0  and  I O Y V   u t  =  1 z  , whic h is exactly F ark as I I(a). No w, the second alternativ e of F ark as I I say s that there is no solution as ab o v e if there is some ( − α, c ) ∈ R d +1 so that ( − α, c ⊤ )  1 z  < 0 a nd ( − α, c ⊤ )  I 0 Y V  ≥ ( O , O ) . These are equiv alen t to − α + c ⊤ z < 0 , − α I + c ⊤ Y ≥ O , c ⊤ V ≥ O , namely , c ⊤ z < α , c ⊤ Y ≥ α I and c ⊤ V ≥ O , whic h are indeed the conditions of F ark as IV(b), in matrix form. Observ e that F a rk as IV can b e view ed as a separation criterion for p olyhedral sets. This v ersion subsumes F ark as I and F ark as I I. 66 CHAPTER 4. POL YHEDRA AND POL YTOPES Chapter 5 Pro jectiv e Spaces, Pro jectiv e P olyhedra, P olar Dualit y w.r.t. a Nondegenerate Quadric 5.1 Pro jectiv e Spaces The fact that not just p oin ts but also v ectors are needed to deal with unbounded p olyhed ra is a hin t that perhaps the notions of p olytop e and p olyhedra can b e uniﬁed by “going pro- jectiv e”. Ho w ev er, we ha v e to b e careful b ecause pro jectiv e g eometry do es not accomo date w ell the notion of conv exit y . This is b ecause con v exit y has to do with con v ex com binations, but the essense of pro jectiv e geometry is that ev erything is deﬁned up to non-zer o scalars, without an y requiremen t that these scalars b e p ositiv e. It is p ossible to dev elop a theory of oriente d pr o je ctive ge ometry (due to J. Stolﬁ [36]) in wic h con ve xit y is nicely accomo dated. Ho w ev er, in this approac h, ev ery p oin t comes as a pair, (p ositiv e p oint, negativ e p oin t), and although it is a v ery elegan t theory , we ﬁnd it a bit un wieldy . Ho w ev er, since all w e really w an t is to “em b ed” E d in to its p r oje ctive c ompletion , P d , so that w e can deal with “p oin ts at inﬁnit y” and “normal p oin t” in a uniform manner in particular, with resp ect to pro jectiv e tra nsformations, w e will con ten t ourselv es with a deﬁnition of the notion of a pro jectiv e p olyhedron using the notion of p olyhedral cone. Th us, w e will not attempt to deﬁne a general notion of con v exit y . W e b egin with a “crash course ” on (real) pro jectiv e spaces. There are man y texts on pro jectiv e geometry . W e suggest starting with G allier [20] and then mo v e on to far more comprehens ive treatmen ts suc h as Berger (Geometry I I) [6] or Sam uel [33]. Deﬁnition 5.1 The (r e al) pr oje ctive sp ac e , RP n , is the set of all lines through the origin in R n +1 , i.e., the set o f one-dimensional subspaces of R n +1 (where n ≥ 0). Since a one- dimensional subspace, L ⊆ R n +1 , is spanned b y an y nonzero vec tor, u ∈ L , w e can view RP n as the set of equ iv alence classes of nonzero v ectors in R n +1 − { 0 } mo dulo the equiv a lence 67 68 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY relation, u ∼ v iﬀ v = λu, for some λ ∈ R , λ 6 = 0 . W e ha v e the pro jection, p : ( R n +1 − { 0 } ) → RP n , give n b y p ( u ) = [ u ] ∼ , the equiv alence class of u mo dulo ∼ . W rite [ u ] (or h u i ) for the line, [ u ] = { λu | λ ∈ R } , deﬁned b y the nonzero vec tor, u . Note that [ u ] ∼ = [ u ] − { 0 } , for eve ry u 6 = 0, so the map [ u ] ∼ 7→ [ u ] is a bijection whic h allows us to iden tify [ u ] ∼ and [ u ]. Th us, w e will use b oth notations in terc hangeably as conv enien t. The pro jectiv e spac e, RP n , is sometimes denoted P ( R n +1 ). Since ev ery line, L , in R n +1 in tersects the sphere S n in t w o an tip o dal po in t s, w e can view RP n as the quotien t of the sphere S n b y iden tiﬁcation of a n tipo dal p oin ts. W e call this the spheric a l mo del of RP n . A more subtle construction consists in considering the (upp er) half-sphere instead of the sphere, where the upp er half-sphere S n + is set of p oin ts on the sphere S n suc h that x n +1 ≥ 0. This time, eve ry line t hro ugh the cen ter in tersects the (upp er) half-sphere in a single p oin t, except on the b oundary of the half-sphere, where it in tersects in tw o an tip odal p o in ts a + and a − . Th us, the pro jectiv e space RP n is the quotien t space obtained from the (upper) half-sphere S n + b y iden tifying antipo dal p oin ts a + and a − on the b oundary of the half-sphere. W e call this mo del of RP n the half-spheric al mo del . When n = 2, w e get a circle. When n = 3, the upp er half-sphere is homeomorphic to a closed disk (sa y , b y orthogonal pro jection on to the xy -plane), and RP 2 is in bijection with a closed disk in whic h a ntipo dal p oin ts on its b oundary (a unit circle) hav e b een iden tiﬁed. This is hard to visualize ! In this mo del of the real pro jectiv e space, pro jectiv e lines are great semicircl es on the upp er half-sphere, with an tip odal p oin ts on the b oundary iden tiﬁed. Boundary p oin ts corresp ond to p oin ts at inﬁnity . By orthogonal pro jection, these great semi circles corresp ond to semiellip ses, with an tip odal p oin ts on the b oundary iden tiﬁed. T ra v eling along suc h a pro jectiv e “line,” when w e reac h a bo undary p oin t, w e “wrap around”! In general, the upp er half-sphere S n + is homeomorphic to the closed unit ball in R n , whose b oundary is the ( n − 1)-sphere S n − 1 . F or example, the pro jectiv e space RP 3 is in bijection with the closed unit ball in R 3 , with an tip o dal points on its b oundary (the sphere S 2 ) iden tiﬁed! Another useful w a y of “visualizing” RP n is to use the h yp erplane , H n +1 ⊆ R n +1 , of equation x n +1 = 1. Observ e that for ev ery line, [ u ], through the origin in R n +1 , if u do es not b elong to the hy p erplane, H n +1 (0) ∼ = R n , of equation x n +1 = 0, then [ u ] in tersects H n +1 is a unique p oin t, namely ,  u 1 u n +1 , . . . , u n u n +1 , 1  , where u = ( u 1 , . . . , u n +1 ). The lines, [ u ], for whic h u n +1 = 0 are “p oin ts at inﬁnit y”. Observ e that the set of lines in H n +1 (0) ∼ = R n is the set of p oin ts of the pro jectiv e space, RP n − 1 , and 5.1. PR OJECTIVE SP A CES 69 so, RP n can b e written as the disjoin t union RP n = R n ∐ RP n − 1 . W e can rep eat the ab o v e analysis on RP n − 1 and so w e can think of RP n as the disjoint union RP n = R n ∐ R n − 1 ∐ · · · ∐ R 1 ∐ R 0 , where R 0 = { 0 } consist o f a single p oint. The ab o v e sho ws that there is a n em b edding, R n ֒ → RP n , giv en by ( u 1 , . . . , u n ) 7→ ( u 1 , . . . , u n , 1). It will also b e v ery useful to use homogeneous co ordinates. Giv en an y p oint, p = [ u ] ∼ ∈ RP n , the set { ( λu 1 , . . . , λu n +1 ) | λ 6 = 0 } is called the set of homo gene ous c o or d i n ates of p . Since u 6 = 0, observ e that for all homog e- neous co ordinates, ( x 1 , . . . , x n +1 ), for p , some x i m ust be non-zero. The traditional notation for the homogeneous co ordinates of a p oint, p = [ u ] ∼ , is ( u 1 : · · · : u n : u n +1 ) . There is a useful bijection b et w een certain kinds o f subsets of R d +1 and subsets of RP d . F o r any subset, S , of R d +1 , let − S = {− u | u ∈ S } . Geometrically , − S is the reﬂexion of S ab out 0. Note that for any nonempt y subse t, S ⊆ R d +1 , with S 6 = { 0 } , the sets S , − S and S ∪ − S a ll induce the same set of p oints in pro jectiv e space, RP d , since p ( S − { 0 } ) = { [ u ] ∼ | u ∈ S − { 0 }} = { [ − u ] ∼ | u ∈ S − { 0 }} = { [ u ] ∼ | u ∈ − S − { 0 }} = p (( − S ) − { 0 } ) = { [ u ] ∼ | u ∈ S − { 0 }} ∪ { [ u ] ∼ | u ∈ ( − S ) − { 0 }} = p (( S ∪ − S ) − { 0 } ) , b ecause [ u ] ∼ = [ − u ] ∼ . Using these facts w e o btain a bijection b et w een subsets of RP d and certain subsets of R d +1 . W e sa y that a set, S ⊆ R d +1 , is symmetric iﬀ S = − S . Ob viously , S ∪ − S is symmetric for any set, S . Sa y that a subset, C ⊆ R d +1 , is a double c one iﬀ for ev ery u ∈ C − { 0 } , the en tire line, [ u ], spanned b y u is con tained in C . Again, we exclude the trivial double cone, C = { 0 } . Th us, ev ery double cone can be view ed as a set o f lines through 0. Note that a double cone is symmetric. Giv en an y nonempty subset, S ⊆ RP d , let v ( S ) ⊆ R d +1 b e the set of v ectors, v ( S ) = [ [ u ] ∼ ∈ S [ u ] ∼ ∪ { 0 } . Note that v ( S ) is a double cone. 70 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY Prop osition 5.1 The m ap, v : S 7→ v ( S ) , fr om the set of nonempty subsets of RP d to the set of no nempty, nontrivial double c ones in R d +1 is a bije ction. Pr o of . W e already noted that v ( S ) is non trivial double cone. Consider the map, ps : S 7→ p ( S ) = { [ u ] ∼ ∈ RP d | u ∈ S − { 0 }} . W e lea v e it as an easy exercise to c hec k that ps ◦ v = id and v ◦ ps = id, whic h shows that v and ps are m utual in v erses. Giv en an y subspace, X ⊆ R n +1 , with dim X = k + 1 ≥ 1 and 0 ≤ k ≤ n , a k -dimension al pr oje ctive subsp ac e of RP n is the image, Y = p ( X − { 0 } ), of X − { 0 } under the pro jection p . W e often write Y = P ( X ) . When k = n − 1, w e sa y that Y is a pr oje ctive hyp erplane or simply a hyp erplane . When k = 1, w e say that Y is a pr oje c tive line or simply a li n e . It is easy to see that ev ery ( pro jectiv e) h yp erplane is the ke rnel (zero set) of some linear equation of the form a 1 x 1 + · · · + a n +1 x n +1 = 0 , where one of the a i is nonzero. Con v ersely , the k ernel of any suc h linear equation is a h yp erplane. F urthermore, giv en a (pro jectiv e) hyperplane, H ⊆ RP n , the linear equation deﬁning H is unique up to a nonzero scalar. F o r any i , with 1 ≤ i ≤ n + 1, the set U i = { ( x 1 : · · · : x n +1 ) ∈ RP n | x i 6 = 0 } is a subset of RP n called an aﬃne p atch of RP n . W e hav e a bijection, ϕ i : U i → R n , b et w een U i and R n giv en by ϕ i : ( x 1 : · · · : x n +1 ) 7→  x 1 x i , . . . , x i − 1 x i , x i +1 x i , . . . , x n +1 x i  . This map is w ell deﬁned b ecause if ( y 1 , . . . , y n +1 ) ∼ ( x 1 , . . . , x n +1 ), that is, ( y 1 , . . . , y n +1 ) = λ ( x 1 , . . . , x n +1 ), with λ 6 = 0, then y j y i = λx j λx i = x j x i (1 ≤ i ≤ n + 1) , since λ 6 = 0 and x i , y i 6 = 0. The in v erse, ψ i : R n → U i ⊆ RP n , of ϕ i is giv en b y ψ i : ( x 1 , · · · , x n ) 7→ ( x 1 : · · · x i − 1 : 1 : x i : · · · : x n ) . Observ e that the bijection, ϕ i , b etw een U i and R n can also b e view ed as the bijection ( x 1 : · · · : x n +1 ) 7→  x 1 x i , . . . , x i − 1 x i , 1 , x i +1 x i , . . . , x n +1 x i  , 5.1. PR OJECTIVE SP A CES 71 b et w een U i and the hy p erplane, H i ⊆ R n +1 , of equ ation x i = 1. W e will mak e hea vy use of these bijections. F or example, for an y subse t, S ⊆ RP n , the “view of S from the patc h U i ”, S ↾ U i , is in bijection with v ( S ) ∩ H i , where v ( S ) is the double cone asso ciated with S (see Prop osition 5.1). The aﬃne patche s, U 1 , . . . , U n +1 , co v er the pro jectiv e space RP n , in the sense that ev ery ( x 1 : · · · : x n +1 ) ∈ RP n b elongs to one o f the U i ’s, as no t all x i = 0. The U i ’s turn out to b e op en subs ets of RP n and they ha v e nonempt y o v erlaps. When w e restrict o urselv es to one of the U i , w e ha v e an “a ﬃne view of RP n from U i ”. In particular, on the aﬃne patc h U n +1 , w e hav e the “standard view” of R n em b edded in to RP n as H n +1 , the hy p erplane of equation x n +1 = 1. The compleme nt, H i (0), of U i in RP n is the (pro jectiv e) hy p erplane of equation x i = 0 (a cop y of RP n − 1 ). With resp ect to the aﬃne patc h, U i , the h yp erplane, H i (0), plays the role of hyp erpla ne (of p oints) at inﬁ nity . F ro m now on, for simplicit y of notation, we will write P n for RP n . W e need to deﬁne pro jectiv e maps. Suc h maps are induced b y linear maps. Deﬁnition 5.2 An y injectiv e linear map, h : R m +1 → R n +1 , induces a map, P ( h ) : P m → P n , deﬁned b y P ( h )([ u ] ∼ ) = [ h ( u )] ∼ and called a pr oje ctive map . When m = n and h is bijectiv e, the map P ( h ) is also bijectiv e and it is called a pr oje ctivity . W e hav e t o ch ec k that t his deﬁnition mak es sense, that is, it is compatible with the equiv alence relation, ∼ . F or this, assume that u ∼ v , that is v = λu, with λ 6 = 0 (of course, u , v 6 = 0). As h is linear, w e get h ( v ) = h ( λu ) = λh ( u ) , that is, h ( u ) ∼ h ( v ), whic h sho ws that [ h ( u )] ∼ do es on dep end on the represen tativ e chose n in the equiv alence class of [ u ] ∼ . It is also easy to c hec k that whenev er tw o linear maps, h 1 and h 2 , induce the same pro jectiv e map, i . e. , if P ( h 1 ) = P ( h 2 ), then there is a nonzero scalar, λ , so that h 2 = λh 1 . Wh y did we r equire h to b e injectiv e? Because if h has a nontriv ial k ernel, then, an y nonzero v ector, u ∈ Ker ( h ), is mapp ed to 0, but as 0 do es not correspond to an y p oin t o f P n , the map P ( h ) is undeﬁned on P (Ker ( h )). In some case, w e allo w pro jectiv e maps induced b y non-injectiv e linear maps h . In this case, P ( h ) is a map whose domain is P n − P (Ker ( h )). An example is the map, σ N : P 3 → P 2 , giv en by ( x 1 : x 2 : x 3 : x 4 ) 7→ ( x 1 : x 2 : x 4 − x 3 ) , whic h is undeﬁned a t the p oin t (0 : 0 : 1 : 1). This map is the “homogenization” of the cen tral pro jection (from the nort h p ole, N = (0 , 0 , 1)) from E 3 on to E 2 . 72 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY  Although a pro jectiv e map, f : P n → P n , is induced b y some linear map, h , the map f is not linear! This is b ecause linear com binations of p oin ts in P n do no t make any sense! Another wa y of deﬁning functions (p ossibly partial) b et w een pro jectiv e space s in v olv es using homogeneous p olynomials. If p 1 ( x 1 , . . . , x m +1 ), . . . , p n +1 ( x 1 , . . . , x m +1 ) are n + 1 homo- geneous p o lynom ials all of the same degree, d , and if these n + 1 p olynomials do not v anish sim ultaneously , then we claim that the function, f , giv en b y f ( x 1 : · · · : x m +1 ) = ( p 1 ( x 1 , . . . , x m +1 ) : · · · : p n +1 ( x 1 , . . . , x m +1 )) is indeed a w ell-deﬁned function f r o m P m to P n . Indeed, if ( y 1 , . . . , y m +1 ) ∼ ( x 1 , . . . , x m +1 ), that is, ( y 1 , . . . , y m +1 ) = λ ( x 1 , . . . , x m +1 ), with λ 6 = 0, as the p i are homogeneous o f degree d , p i ( y 1 , . . . , y m +1 ) = p i ( λx 1 , . . . , λx m +1 ) = λ d p i ( x 1 , . . . , x m +1 ) , and so, f ( y 1 : · · · : y m +1 ) = ( p 1 ( y 1 , . . . , y m +1 ) : · · · : p n +1 ( y 1 , . . . , y m +1 )) = ( λ d p 1 ( x 1 , . . . , x m +1 ) : · · · : λ d p n +1 ( x 1 , . . . , x m +1 )) = λ d ( p 1 ( x 1 , . . . , x m +1 ) : · · · : p n +1 ( x 1 , . . . , x m +1 )) = λ d f ( x 1 : · · · : x m +1 ) , whic h sho ws that f ( y 1 : · · · : y m +1 ) ∼ f ( x 1 : · · · : x m +1 ), as required. F o r example, the map, τ N : P 2 → P 3 , giv en b y ( x 1 : x 2 , : x 3 ) 7→ (2 x 1 x 3 : 2 x 2 x 3 : x 2 1 + x 2 2 − x 2 3 : x 2 1 + x 2 2 + x 2 3 ) , is w ell-deﬁned. It turns out to b e the “homogenization” of the in v erse stere ogra phic map from E 2 to S 2 (see Section 8.5). Observ e that τ N ( x 1 : x 2 : 0) = (0 : 0 : x 2 1 + x 2 2 : x 2 1 + x 2 2 ) = (0 : 0 : 1 : 1) , that is, τ N maps all the p oin ts at inﬁnit y (in H 3 (0)) to the “north p ole”, (0 : 0 : 1 : 1). Ho w ev er, when x 3 6 = 0, w e can prov e that τ N is injectiv e (in fact, its inv erse is σ N , deﬁned earlier). Most in teresting subsets of pro jectiv e space a r ise as the collection of zeros of a (ﬁnite) set of homogeneous p olynomials. Let us b egin with a single homogeneous p olynomial, p ( x 1 , . . . , x n +1 ), of degree d and set V ( p ) = { ( x 1 : · · · : x n +1 ) ∈ P n | p ( x 1 , . . . , x n +1 ) = 0 } . As usual, we need to c hec k that this deﬁnition do es not dep end on the sp eciﬁc represen tativ e c hosen in the equiv alence class of [( x 1 , . . . , x n +1 )] ∼ . If ( y 1 , . . . , y n +1 ) ∼ ( x 1 , . . . , x n +1 ), that is, ( y 1 , . . . , y n +1 ) = λ ( x 1 , . . . , x n +1 ), with λ 6 = 0, as p is homogeneous of degree d , p ( y 1 , . . . , y n +1 ) = p ( λx 1 , . . . , λx n +1 ) = λ d p ( x 1 , . . . , x n +1 ) , 5.1. PR OJECTIVE SP A CES 73 and as λ 6 = 0, p ( y 1 , . . . , y n +1 ) = 0 iﬀ p ( x 1 , . . . , x n +1 ) = 0 , whic h sho ws t hat V ( p ) is well deﬁned. F or a set of homogeneous p olynomials ( not neces sarily of the same degree), E = { p 1 ( x 1 , . . . , x n +1 ) , . . . , p s ( x 1 , . . . , x n +1 ) } , w e set V ( E ) = s \ i =1 V ( p i ) = { ( x 1 : · · · : x n +1 ) ∈ P n | p i ( x 1 , . . . , x n +1 ) = 0 , i = 1 . . . , s } . The set, V ( E ), is usually called the pr oje ctive variety deﬁned b y E (or cut out by E ). When E consists of a single polynomial, p , the set V ( p ) is called a (pro j ectiv e) hyp ersurfac e . F or example, if p ( x 1 , x 2 , x 3 , x 4 ) = x 2 1 + x 2 2 + x 2 3 − x 2 4 , then V ( p ) is the pr oje ctive spher e in P 3 , also denote d S 2 . Indeed, if w e “lo ok” at V ( p ) o n the aﬃne patch U 4 , where x 4 6 = 0, w e kno w that t his amoun ts to setting x 4 = 1, and we do get the set of p o in ts ( x 1 , x 2 , x 3 , 1) ∈ U 4 satisfying x 2 1 + x 2 2 + x 2 3 − 1 = 0, our usual 2-sphere! Ho w ev er, if w e lo ok at V ( p ) on the patc h U 1 , where x 1 6 = 0, w e see the quadric of equation 1 + x 2 2 + x 2 3 = x 2 4 , whic h is not a sphere but a h yp erboloid o f tw o sheets ! Nev ertheless, if w e pic k x 4 = 0 as the plane at inﬁnit y , note that the pro jectiv e sphere do es not ha v e p oints at inﬁnit y since the only r e al solution of x 2 1 + x 2 2 + x 2 3 = 0 is (0 , 0 , 0), but (0 , 0 , 0 , 0) do es not correspo nd to an y p oin t of P 3 . Another example is giv en b y q = ( x 1 , x 2 , x 3 , x 4 ) = x 2 1 + x 2 2 − x 3 x 4 , for whic h V ( q ) corresponds to a parab oloid in the patc h U 4 . Indeed, if w e set x 4 = 1, we get the set of p oin ts in U 4 satisfying x 3 = x 2 1 + x 2 2 . F or this reason, we denote V ( q ) by P a nd called it a (pr oje ctive) p ar ab oloi d . Giv en an y homogeneous p olynomial, F ( x 1 , . . . , x d +1 ), w e will also mak e use of the h yp er- surfac e c one , C ( F ) ⊆ R d +1 , deﬁned b y C ( F ) = { ( x 1 , . . . , x d +1 ) ∈ R d +1 | F ( x 1 , . . . , x d +1 ) = 0 } . Observ e that V ( F ) = P ( C ( F )). Remark: Ev ery v ariet y , V ( E ), deﬁned b y a set of p olynomials, E = { p 1 ( x 1 , . . . , x n +1 ) , . . . , p s ( x 1 , . . . , x n +1 ) } , is a lso the h ypersurface deﬁned b y the singl e p olynomial equation, p 2 1 + · · · + p 2 s = 0 . This fact, p eculiar to the real ﬁeld, R , is a mixed blessing. On the one-hand, the study of v arieties is reduced to the study of h yp ersurfaces . On the other-hand, this is a hint that w e should expect that suc h a study will b e hard. 74 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY P erhaps to the surprise of the no vice, there is a bijectiv e pro j ectiv e map (a pro j ecti vity ) sending S 2 to P . This map, θ , is given b y θ ( x 1 : x 2 : x 3 : x 4 ) = ( x 1 : x 2 : x 3 + x 4 : x 4 − x 3 ) , whose in v erse is g iven b y θ − 1 ( x 1 : x 2 : x 3 : x 4 ) =  x 1 : x 2 : x 3 − x 4 2 : x 3 + x 4 2  . Indeed, if ( x 1 : x 2 : x 3 : x 4 ) satisﬁes x 2 1 + x 2 2 + x 2 3 − x 2 4 = 0 , and if ( z 1 : z 2 : z 3 : z 4 ) = θ ( x 1 : x 2 : x 3 : x 4 ), then from ab o v e, ( x 1 : x 2 : x 3 : x 4 ) =  z 1 : z 2 : z 3 − z 4 2 : z 3 + z 4 2  , and b y plugging the right-hand sides in the equation of the sphere, w e get z 2 1 + z 2 2 +  z 3 − z 4 2  2 −  z 3 + z 4 2  2 = z 2 1 + z 2 2 + 1 4 ( z 2 3 + z 2 4 − 2 z 3 z 4 − ( z 2 3 + z 2 4 + 2 z 3 z 4 )) = z 2 1 + z 2 2 − z 3 z 4 = 0 , whic h is the equation of the parab oloid, P . 5.2 Pro jectiv e P o l yhedra F o llowing the “pro jectiv e do ctrine” whic h consists in replacing p oin ts b y lines through the origin, that is, to “conify” ev erything, w e will deﬁne a pro jectiv e p olyhed ron as an y set o f p oin ts in P d induced by a p olyhedral cone in R d +1 . T o do so, it is preferable to consider cones as sets of p ositiv e combinations of v ectors (see Deﬁnition 4.3). Just to refresh our memory , a set, C ⊆ R d , is a V -c one or p olyhe d r al c one if C is the p ositiv e h ull of a ﬁnite set of v ectors, that is, C = cone( { u 1 , . . . , u p } ) , for some v ectors, u 1 , . . . , u p ∈ R d . An H -c one is any subset of R d giv en by a ﬁnite in tersection of closed half-spaces cut out b y h yp erplanes through 0. A go o d place to learn ab out cones (and m uc h more) is F ulton [19]. See also Ew ald [18]. By Theorem 4.18, V -cones and H -cones form the same collection of con v ex sets (for ev ery d ≥ 0). Naturally , we can think of these cones as sets o f rays (half-lines) of the form h u i + = { λu | λ ∈ R , λ ≥ 0 } , 5.2. PR OJECTIVE POL YHEDRA 75 where u ∈ R d is any nonzer o v ector. W e exclude the trivial cone, { 0 } , since 0 do es not deﬁne an y p oin t in pro jectiv e space. When w e “g o pro jectiv e”, eac h ray corresp onds to the full line, h u i , spanned b y u whic h can b e expresse d as h u i = h u i + ∪ −h u i + , where −h u i + = h u i − = { λu | λ ∈ R , λ ≤ 0 } . No w, if C ⊆ R d is a p olyhedral cone, ob viously − C is also a p olyhedral cone and the set C ∪ − C consists of the union of the tw o p olyhedral cones C and − C . Note that C ∪ − C can b e view ed as the set of all lines determined b y the nonzero vec tors in C (and − C ). It is a double cone. Unless C is a closed half-space, C ∪ − C is not conv ex. It seems p erfectly natural to deﬁne a pro jectiv e p olyhedron a s an y set of lines induced b y a set of the form C ∪ − C , where C is a po lyhedral cone. Deﬁnition 5.3 A pr oje ctive p olyhe dr on is an y subset, P ⊆ P d , of the form P = p (( C ∪ − C ) − { 0 } ) = p ( C − { 0 } ) , where C is an y p olyhedral cone ( V or H cone) in R d +1 (with C 6 = { 0 } ). W e write P = P ( C ∪ − C ) or P = P ( C ). It is imp ortan t to observ e that b ecause C ∪− C is a double cone there is a bijection b et we en non trivial double p olyhedral cones and pro jectiv e p olyhedra. So, pro jectiv e p olyhedra are equiv alen t to double p olyhed ral cones . Ho w ev er, the pro jectiv e in terpretation of the lines induced b y C ∪ − C as p oin ts in P d mak es the study of pro jectiv e p olyhedra geometrically more in teresting. Pro jectiv e p olyhedra inherit man y of the properties of cones but w e ha v e to b e careful b ecause w e are really dealing with double cones, C ∪ − C , and not cones. As a conseq uence, there are a few unpleasan t surprises, for example, the fact that the collection of pro jectiv e p olyhedra is not close d under in tersection! Before dealing with these issue s, let us sho w that eve ry “standard” p olyhe dron, P ⊆ E d , has a nat ura l pro jectiv e completion, e P ⊆ P d , suc h that on the aﬃne patc h U d +1 (where x d +1 6 = 0), e P ↾ U d +1 = P . F or this, w e use our theorem on the P olyhedron–Cone Corresp ondence (Theorem 4.19, part (2)). Let A = X + U , where X is a set of p oints in E d and U is a cone in R d . F or ev ery p oin t, x ∈ X , and ev ery vec tor, u ∈ U , let b x =  x 1  , b u =  u 0  , and let b X = { b x | x ∈ X } and b U = { b u | u ∈ U } . Then, C ( A ) = cone( { b X ∪ b U } ) 76 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY is a cone in R d +1 suc h that b A = C ( A ) ∩ H d +1 , where H d +1 is the h yp erplane of eq uation x d +1 = 1. If w e set e A = P ( C ( A )), then w e get a subset of P d and in the patc h U d +1 , the set e A ↾ U d +1 is in bijection with the in tersection ( C ( A ) ∪ − C ( A )) ∩ H d +1 = b A , and th us, in bijection with A . W e call e A the pr oje ctive c om pletion of A . W e hav e an injection, A − → e A , giv en b y ( a 1 , . . . , a d ) 7→ ( a 1 : · · · : a d : 1) , whic h is just the map, ψ d +1 : R d → U d +1 . What the pro jectiv e completion do es is to add to A the “p oin ts at inﬁnit y” corresp onding to the v ectors in U , that is, the p oin ts o f P d correspo nding to the lines in the cone, U . In particular, if X = conv( Y ) and U = cone( V ), for some ﬁnite sets Y = { y 1 , . . . , y p } and V = { v 1 , . . . , v q } , then P = conv( Y ) + cone( V ) is a V -p olyhedron and e P = P ( C ( P )) is a pro jectiv e p olyhe dron. The pro jectiv e p olyhedron, e P = P ( C ( P )), is called the pr oj e ctive c ompletion of P . Observ e that if C is a closed half-space in R d +1 , then P = P ( C ∪ − C ) = P d . Now , if C ⊆ R d +1 is a p olyhedral cone and C is contained in a closed half-space, it is still p ossible that C con tains some non trivial linear subspace and we w ould lik e to understand this situation. The ﬁrst thing to observ e is that U = C ∩ ( − C ) is the largest linear subspace con tained in C . If C ∩ ( − C ) = { 0 } , w e sa y that C is a p ointe d or str ongly c onvex cone. In this case, one immediately realizes that 0 is an extreme p oin t of C and so, there is a h yp erplane, H , through 0 so that C ∩ H = { 0 } , that is, except for its ap ex, C lies in one of the open half-spaces determined b y H . As a conseque nce, by a linear c hange of co ordinates, w e may assume that this hyperplane is H d +1 and so, for ev ery pro jectiv e p olyhedron, P = P ( C ), if C is p oin ted then there is an aﬃne patc h (sa y , U d +1 ) where P has no p oin ts at inﬁnit y , that is, P is a p olytop e! On the other hand, fro m another patc h, U i , as P ↾ U i is in bijection with ( C ∪ − C ) ∩ H i , the pro jectiv e p o lyhe dron P view ed on U i ma y consis t of two disjoin t p olyhedra. The situation is v ery similar to the classical theory of pro jectiv e conics or quadrics (for example, see Brannan, Esplen and G ra y , [10]). The case where C is a p oin ted cone corre- sp onds to the nondegenerate conics or quadrics . In the case of the conics, depending ho w w e slice a cone, w e see a n ellipse, a parab ola or a h yp erb ola. F or pro jectiv e p olyhed ra, when w e slice a p olyhedral double cone, C ∪ − C , w e ma y see a p olytop e ( el liptic typ e ) a single un b ounded p olyhedron ( p ar ab olic typ e ) or t w o un b ounded p olyhedra ( hyp erb olic typ e ). No w, when U = C ∩ ( − C ) 6 = { 0 } , the p olyhed ral cone, C , con tains the linear subspace, U , and if C 6 = R d +1 , then for ev ery h yp erplane, H , suc h that C is con tained in one of the tw o closed half- space s determined b y H , the subspace U ∩ H is no ntrivial. An examp le is the cone, C ⊆ R 3 , determined by the interse ction of t w o planes through 0 (a w edge). In this case, U is equal to the line of inters ection of these tw o planes . Also observ e that C ∩ ( − C ) = C iﬀ C = − C , that is, iﬀ C is a linear subspace. 5.2. PR OJECTIVE POL YHEDRA 77 The situation where C ∩ ( − C ) 6 = { 0 } is reminisc en t of the case of cylinde rs in the theory of quadric surfaces ( see [10] or Berger [6]). Now, ev ery cylinder can b e vie we d as the r uled surface deﬁned as the family of lines orthogonal to a plane and touc hing some nondegenerate conic. A similar decomposition holds for p olyhedral cones as sho wn b elo w in a prop osition b orro w ed fr o m Ew a ld [18] (Chapter V, Lemma 1.6). W e should w arn the reader that w e ha v e some doubts ab out the pro of giv en there and so, we oﬀer a diﬀeren t pro of adapted fro m the pro of of Lemma 16.2 in Ba rvinok [3]. Giv en an y tw o subsets, V , W ⊆ R d , as usual, w e write V + W = { v + w | v ∈ V , w ∈ W } and v + W = { v + w | w ∈ W } , for any v ∈ R d . Prop osition 5.2 F or eve ry p olyhe dr al c one, C ⊆ R d , if U = C ∩ ( − C ) , then ther e is s o me p oi n te d c one, C 0 , so that U and C 0 ar e ortho gonal a nd C = U + C 0 , with dim( U ) + dim( C 0 ) = dim( C ) . Pr o of . W e already kno w that U = C ∩ ( − C ) is the largest linear subspace of C . Let U ⊥ b e the orthog o nal complem ent of U in R d and let π : R d → U ⊥ b e the orthogonal pro jection on to U ⊥ . By Prop osition 4.12, the pro jection, C 0 = π ( C ), of C onto U ⊥ is a polyhedral cone. W e claim that C 0 is p oin ted and that C = U + C 0 . Since π − 1 ( v ) = v + U for ev ery v ∈ C 0 , w e ha v e U + C 0 ⊆ C . On the other hand, b y deﬁnition of C 0 , we a lso hav e C ⊆ U + C 0 , so C = U + C 0 . If C 0 w as not p oin ted, then it w ould contain a linear subspace, V , of dimension at least 1 but then, U + V would b e a linear subspace of C of dimension strictly g reater than U , whic h is imp ossible. Finally , dim( U ) + dim( C 0 ) = dim( C ) is ob vious b y orthogo nality . The linear subspace, U = C ∩ ( − C ), is called the c osp an of C . Both U a nd C 0 are uniquely determined b y C . T o a g reat exten t, Prop osition reduces the study of non-p o in ted cones to the study of p oin ted cones. W e prop ose to call the pro jectiv e p olyhedra o f the form P = P ( C ), whe re C is a cone with a non-trivial cospan (a non-p oin ted cone) a pr oje ctive p ol yhe dr al cylinder , by analogy with the quadric surfaces. W e also prop ose to call the pro jectiv e p olyhedra of the form P = P ( C ), where C is a p oin ted cone, a pr oje ctive p olytop e (or nonde gener ate pr oje ctive p olyhe dr on ). The following prop ositions sho w that pro jectiv e po lyhe dra b eha v e well under pro jectiv e maps and in tersection with a h yp erplane: Prop osition 5.3 Given any pr oje ctive map, h : P m → P n , for any pr oje ctive p o lyhe dr on, P ⊆ P m , the image, h ( P ) , o f P is a pr oje ctive p olyhe dr on in P n . Even if h : P m → P n is a p artial map but h is deﬁne d on P , then h ( P ) is a pr oje ctive p olyhe d r on. 78 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY Pr o of . The pro jectiv e map, h : P m → P n , is of the form h = P ( b h ), for some injectiv e linear map, b h : R m +1 → R n +1 . Moreo v er, the pro jectiv e p olyhedron, P , is of the form P = P ( C ), for some p olyhedral cone, C ⊆ R n +1 , with C = cone( { u 1 , . . . , u p } ), for some nonzero v ector u i ∈ R d +1 . By deﬁnition, P ( h )( P ) = P ( h )( P ( C )) = P ( b h ( C )) . As b h is linear, b h ( C ) = b h (cone( { u 1 , . . . , u p } )) = cone( { b h ( u 1 ) , . . . , b h ( u p ) } ) . If w e let b C = cone( { b h ( u 1 ) , . . . , b h ( u p ) } ), then b h ( C ) = b C is a p olyhedral cone and so, P ( h )( P ) = P ( b h ( C )) = P ( b C ) is a pro jectiv e cone. This argumen t do es not dep end on the injectivit y of b h , as long as C ∩ Ker ( b h ) = { 0 } . Prop osition 5.3 together with earlier argumen ts show s that ev ery pro jectiv e p olytop e, P ⊆ P d , is equiv alen t under some suitable pro jectivit y to another pro jectiv e p olytope, P ′ , whic h is a p olytop e when view ed in the aﬃne patch, U d +1 . This prop ert y is similar to the fact that ev ery (non-degenerate) pro jectiv e conic is pro jectiv ely equ iv alen t to an ellips e. Since the notion of a face is deﬁned for arbitrary p olyhed ra it is a lso deﬁned for cones . Conseque ntly , we can deﬁne the notion of a face for pro jectiv e p olyhedra. Giv en a pro jectiv e p olyhedron, P ⊆ P d , where P = P ( C ) for some p olyhedral cone (uniquely determined by P ), C ⊆ R d +1 , a fac e of P is any subset of P of the form P ( F ) = p ( F − { 0 } ), for an y non trivial face, F ⊆ C , of C ( F 6 = { 0 } ). Consequen tly , w e sa y that P ( F ) is a v e rtex iﬀ dim ( F ) = 1, an e dge iﬀ dim( F ) = 2 and a fac et iﬀ dim( F ) = dim( C ) − 1. The pro jectiv e p olyhedron, P , and the empt y set are the impr op er faces of P . If C is strongly con v ex, then it is easy to pro v e that C is generated b y its edges (its one-dimensional faces, these are rays) in the sense that an y set of nonzero v ector spanning these edges generate C (using p ositiv e linear com binations). As a conseq uence, if C is strongly conv ex, we ma y say that P is “spanned” b y its v ertices, since P is equal to P (all p ositiv e com binations of v ectors r epresen ting its v ertices). Remark: Ev en though w e did not deﬁne the notio n of con v ex com bination of points in P d , the notion of pro jectiv e p olyhedron giv es us a w ay to mimic certain prop erties of con ve x sets in the framew ork of pro jectiv e geometry . That’s because ev ery pro jectiv e p olyhedron correspo nds to a unique p olyhedral cone. If our pro jectiv e p olyhedron is the completion, e P = P ( C ( P )) ⊆ P d , of some p olyhedron, P ⊆ R d , then eac h face o f the cone, C ( P ), is o f the form C ( F ), where F is a face of P and so, eac h face of e P is of the form P ( C ( F )), for some face, F , of P . In particular, in the aﬃne patc h, U d +1 , the face, P ( C ( F ) ), is in bijection with the face, F , of P . W e will usually iden tify P ( C ( F )) and F . 5.2. PR OJECTIVE POL YHEDRA 79 W e now consider the in tersection of pro jectiv e p olyhedra but ﬁrst, let us mak e some general remarks ab out the in tersection of subsets of P d . Given an y t w o nonempt y subsets , P ( S ) and P ( S ′ ), of P d what is P ( S ) ∩ P ( S ′ )? It is tempting to sa y that P ( S ) ∩ P ( S ′ ) = P ( S ∩ S ′ ) , but unfortunately this is generally false! The problem is that P ( S ) ∩ P ( S ′ ) is the set of al l lines determined by v ectors b oth in S and S ′ but there ma y be some line spanned b y some v ector u ∈ ( − S ) ∩ S ′ or u ∈ S ∩ ( − S ′ ) suc h that u do es not b elong t o S ∩ S ′ or − ( S ∩ S ′ ). Observ e that − ( − S ) = S − ( S ∩ S ′ ) = ( − S ) ∩ ( − S ′ ) . Then, the correct in tersection is giv en by ( S ∪ − S ) ∩ ( S ′ ∪ − S ′ ) = ( S ∩ S ′ ) ∪ (( − S ) ∩ ( − S ′ )) ∪ ( S ∩ ( − S ′ )) ∪ (( − S ) ∩ S ′ ) = ( S ∩ S ′ ) ∪ − ( S ∩ S ′ ) ∪ ( S ∩ ( − S ′ )) ∪ − ( S ∩ ( − S ′ )) , whic h is the union of t w o double cones (except for 0, which belongs to bot h). Therefore, P ( S ) ∩ P ( S ′ ) = P ( S ∩ S ′ ) ∪ P ( S ∩ ( − S ′ )) = P ( S ∩ S ′ ) ∪ P (( − S ) ∩ S ′ ) , since P ( S ∩ ( − S ′ )) = P (( − S ) ∩ S ′ ). F urthermore, if S ′ is symmetric ( i.e. , S ′ = − S ′ ), then ( S ∪ − S ) ∩ ( S ′ ∪ − S ′ ) = ( S ∪ − S ) ∩ S ′ = ( S ∩ S ′ ) ∪ (( − S ) ∩ S ′ ) = ( S ∩ S ′ ) ∪ − ( S ∩ ( − S ′ )) = ( S ∩ S ′ ) ∪ − ( S ∩ S ′ ) . Th us, if either S or S ′ is symmetric , it is true that P ( S ) ∩ P ( S ′ ) = P ( S ∩ S ′ ) . No w, if C is a p oin ted polyhedral cone then C ∩ ( − C ) = { 0 } . Consequen tly , for a n y other p olyhedral cone, C ′ , we ha v e ( C ∩ C ′ ) ∩ (( − C ) ∩ C ′ ) = { 0 } . Using these facts w e obtain the follo wing result: Prop osition 5.4 L et P = P ( C ) and P ′ = P ( C ′ ) b e any two pr oje ctive p olyhe dr a in P d . If P ( C ) ∩ P ( C ′ ) 6 = ∅ , then the fol lowing pr op erties hold: (1) P ( C ) ∩ P ( C ′ ) = P ( C ∩ C ′ ) ∪ P ( C ∩ ( − C ′ )) , the union of two pr oje ctive p olyhe dr a. If C or C ′ is a p ointe d c on e i.e., P or P ′ is a pr oje ctive p olytop e, then P ( C ∩ C ′ ) and P ( C ∩ ( − C ′ )) ar e disjoint. 80 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY (2) If P ′ = H , for some hyp erplane, H ⊆ P d , then P ∩ H is a pr oje c tive p o lyhe dr on. Pr o of . W e already prov ed (1) so only (2) remains to b e pro v ed. Of cou rse, we ma y assume that P 6 = P d . This time, using the equiv alence theorem of V - cones a nd H -cones (Theorem 4.18), w e kno w that P is of the form P = P ( C ), with C = T p i =1 C i , where the C i are closed half-spaces in R d +1 . Moreo v er, H = P ( b H ), for some h yp erplane, b H ⊆ R d +1 , through 0. Now, as b H is symmetric, P ∩ H = P ( C ) ∩ P ( b H ) = P ( C ∩ b H ) . Conseque ntly , P ∩ H = P ( C ∩ b H ) = P p \ i =1 C i ! ∩ b H ! . Ho w ev er, b H = b H + ∩ b H − , where b H + and b H − are the t w o closed half-spaces determined b y b H and so, b C = p \ i =1 C i ! ∩ b H = p \ i =1 C i ! ∩ b H + ∩ b H − is a p olyhe dral cone. Therefore, P ∩ H = P ( b C ) is a pro jectiv e p olyhedron. W e lea v e it as a n instructiv e exerc ise to ﬁnd explicit examples where P ∩ P ′ consists of t w o disjoin t pro jectiv e p olyhedra in P 1 (or P 2 ). Prop osition 5.4 can be sharp ened a little. Prop osition 5.5 L et P = P ( C ) and P ′ = P ( C ′ ) b e any two pr oje ctive p olyhe dr a in P d . If P ( C ) ∩ P ( C ′ ) 6 = ∅ , then P ( C ) ∩ P ( C ′ ) = P ( C ∩ C ′ ) ∪ P ( C ∩ ( − C ′ )) , the union of two pr oje ctive p olyhe dr a. If C = − C , i.e., C is a line ar subsp ac e (or if C ′ is a line ar subsp ac e), then P ( C ) ∩ P ( C ′ ) = P ( C ∩ C ′ ) . F urthermor e, if either C or C ′ is p ointe d, the a b ove pr oje ctive p olyhe dr a ar e disjoi n t, else if C and C ′ b oth have nontrivial c osp an an d P ( C ∩ C ′ ) and P ( C ∩ ( − C ′ )) interse ct then P ( C ∩ C ′ ) ∩ P ( C ∩ ( − C ′ )) = P ( C ∩ ( C ′ ∩ ( − C ′ ))) ∪ P ( C ′ ∩ ( C ∩ ( − C ))) . Final ly, if the two pr oje ctive p o l yhe dr a o n the right-hand sid e interse ct, then P ( C ∩ ( C ′ ∩ ( − C ′ ))) ∩ P ( C ′ ∩ ( C ∩ ( − C ))) = P (( C ∩ ( − C )) ∩ ( C ′ ∩ ( − C ′ ))) . Pr o of . Left as a simpl e exerc ise in b o olean algebra. In preparation for Section 8 .6 , we also need the notion of tangen t space at a po int of a v ariety . 5.3. T ANGENT SP A CES OF HYPERSURF ACE S 81 5.3 T angen t Spaces of Hyp ersurfaces and Pro jectiv e Hyp ersurfaces Since we only need to consider the case of h yp ersurfaces w e restrict attention to this case (but the general case is a straightforw ard generalization). Let us begin with a h ypersurface of equation p ( x 1 , . . . , x d ) = 0 in R d , that is, the set S = V ( p ) = { ( x 1 , . . . , x d ) ∈ R d | p ( x 1 , . . . , x d ) = 0 } , where p ( x 1 , . . . , x d ) is a p olynomial of total degree, m . Pic k any p oin t a = ( a 1 , . . . , a d ) ∈ R d . Recall that there is a v ersion of the T aylor expansion formula fo r p olynomials suc h that, for any p olynomial, p ( x 1 , . . . , x d ), of total degree m , for ev ery h = ( h 1 , . . . , h d ) ∈ R d , w e ha v e p ( a + h ) = p ( a ) + X 1 ≤| α |≤ m D α p ( a ) α ! h α = p ( a ) + d X i =1 p x i ( a ) h i + X 2 ≤| α |≤ m D α p ( a ) α ! h α , where w e use the multi-index n o tation , with α = ( i 1 , . . . , i d ) ∈ N d , | α | = i 1 + · · · + i d , α ! = i 1 ! · · · i d !, h α = h i 1 1 · · · h i d d , D α p ( a ) = ∂ i 1 ∂ x i 1 1 · · · ∂ i d p ∂ x i d d ( a ) , and p x i ( a ) = ∂ p ∂ x i ( a ) . Consider an y line, ℓ , through a , giv en pa ra metric ally by ℓ = { a + th | t ∈ R } , with h 6 = 0 and sa y a ∈ S is a p oint on the h ypersurface, S = V ( p ), whic h means that p ( a ) = 0. The in tuitiv e idea b ehind the notion of the tangen t space to S at a is that it is the set o f lines that in tersect S at a in a p o in t of multiplicity at le ast two , whic h means that the equation giving the in tersection, S ∩ ℓ , namely p ( a + th ) = p ( a 1 + th 1 , . . . , a d + th d ) = 0 , is of the form t 2 q ( a, h )( t ) = 0 , 82 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY where q ( a, h )( t ) is some p olynomial in t . Using T a ylor’s formula, as p ( a ) = 0, w e ha v e p ( a + th ) = t d X i =1 p x i ( a ) h i + t 2 q ( a, h )( t ) , for some p olynomial, q ( a, h )( t ). F rom this, we see that a is an in tersection p oin t of m ulti- plicit y at least 2 iﬀ d X i =1 p x i ( a ) h i = 0 . ( † ) Conseque ntly , if ∇ p ( a ) = ( p x 1 ( a ) , . . . , p x d ( a )) 6 = 0 (that is, if the gradien t o f p at a is nonzero), w e see that ℓ in tersects S at a in a p oint of multip licit y at least 2 iﬀ h b elongs to the h yp erplane of equation ( † ). Deﬁnition 5.4 Let S = V ( p ) b e a h yp ersurface in R d . F or an y p oin t, a ∈ S , if ∇ p ( a ) 6 = 0, then w e sa y that a is a non-singular point of S . When a is nonsingular, the ( aﬃne ) tangent sp a c e , T a ( S ) (or simply , T a S ), to S at a is the hy p erplane through a of equation d X i =1 p x i ( a )( x i − a i ) = 0 . Observ e that the h yperplane of t he direction of T a S is the h yp erplane through 0 and parallel to T a S giv en b y d X i =1 p x i ( a ) x i = 0 . When ∇ p ( a ) = 0, w e either sa y that T a S is undeﬁned or we set T a S = R d . W e no w extend the notion of tangen t space to pro jectiv e v arieties. As w e will see, this amoun ts to homogenizing and the result turns out to b e simple r than the aﬃne case! So, let S = V ( F ) ⊆ P d b e a pro jectiv e h yp ersurface, whic h means that S = V ( F ) = { ( x 1 : · · · : x d +1 ) ∈ P d | F ( x 1 , . . . , x d +1 ) = 0 } , where F ( x 1 , . . . , x d +1 ) is a homogeneous p olynomial of total degree, m . Again, we sa y that a p oint, a ∈ S , is non-singular iﬀ ∇ F ( a ) = ( F x 1 ( a ) , . . . , F x d +1 ( a )) 6 = 0. F or ev ery i = 1 , . . . , d + 1, let z j = x j x i , where j = 1 , . . . , d + 1 and j 6 = i , and let f ↾ i b e the result of “dehomogenizing” F at i , that is, f ↾ i ( z 1 , . . . , z i − 1 , z i +1 , . . . , z d +1 ) = F ( z 1 , . . . , z i − 1 , 1 , z i +1 , . . . , z d +1 ) . 5.3. T ANGENT SP A CES OF HYPERSURF ACE S 83 W e deﬁne the (pr oje ctive) tangent sp ac e , T a S , to a at S as the h yp erplane, H , suc h that for eac h aﬃne patch , U i where a i 6 = 0, if w e let a ↾ i j = a j a i , where j = 1 , . . . , d + 1 and j 6 = i , then the restriction, H ↾ U i , of H to U i is the aﬃne h yp erplane tangen t to S ↾ U i giv en by d +1 X j =1 j 6 = i f ↾ i z j ( a ↾ i )( z j − a ↾ i j ) = 0 . Th us, on the aﬃne patc h, U i , the tangent space, T a S , is giv en by the homogeneous equation d +1 X j =1 j 6 = i f ↾ i z j ( a ↾ i )( x j − a ↾ i j x i ) = 0 . This lo oks awful but w e can mak e it prett y if w e remem b er that F is a homogeneous p oly- nomial of degree m and that w e ha v e the Euler r elation : d +1 X j =1 F x j ( x ) x j = mF , for ev ery x = ( x 1 , . . . , x d +1 ) ∈ R d +1 . Using this, w e can come up with a clean equation for our pro jectiv e tangen t h yp erplane. It is enough to carry out the computations for i = d + 1. Our tangen t h yp erplane has the equation d X j =1 F x j ( a ↾ d +1 1 , . . . , a ↾ d +1 d , 1)( x j − a ↾ d +1 j x d +1 ) = 0 , that is, d X j =1 F x j ( a ↾ d +1 1 , . . . , a ↾ d +1 d , 1) x j + d X j =1 F x j ( a ↾ d +1 1 , . . . , a ↾ d +1 d , 1)( − a ↾ d +1 j x d +1 ) = 0 . As F ( x 1 , . . . , x d +1 ) is homogeneous of degree m , and as a d +1 6 = 0 on U d +1 , w e ha v e a m d +1 F ( a ↾ d +1 1 , . . . , a ↾ d +1 d , 1) = F ( a 1 , . . . , a d , a d +1 ) , so from the ab o v e equation w e get d X j =1 F x j ( a 1 , . . . , a d +1 ) x j + d X j =1 F x j ( a 1 , . . . , a d +1 )( − a ↾ d +1 j x d +1 ) = 0 . ( ∗ ) 84 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY Since a ∈ S , we ha v e F ( a ) = 0, so the Euler relation yields d X j =1 F x j ( a 1 , . . . , a d +1 ) a j + F x d +1 ( a 1 , . . . , a d +1 ) a d +1 = 0 , whic h, b y dividing by a d +1 and m ultiplying b y x d +1 , yields d X j =1 F x j ( a 1 , . . . , a d +1 )( − a ↾ d +1 j x d +1 ) = F x d +1 ( a 1 , . . . , a d +1 ) x d +1 , and b y plugging this in ( ∗ ), w e get d X j =1 F x j ( a 1 , . . . , a d +1 ) x j + F x d +1 ( a 1 , . . . , a d +1 ) x d +1 = 0 . Conseque ntly , the tangen t h yp erplane to S at a is given by the equation d +1 X j =1 F x j ( a ) x j = 0 . Deﬁnition 5.5 Let S = V ( F ) b e a h yp ersurface in P d , where F ( x 1 , . . . , x d +1 ) is a homoge- neous p olynomial. F or an y p oin t, a ∈ S , if ∇ F ( a ) 6 = 0, then w e sa y that a is a non-singular p oin t of S . When a is nonsingular, the ( pr oje ctive ) tangent sp ac e , T a ( S ) (or simply , T a S ), to S at a is the h yperplane through a of equation d +1 X i =1 F x i ( a ) x i = 0 . F o r example, if we consider the sphere, S 2 ⊆ P 3 , of equation x 2 + y 2 + z 2 − w 2 = 0 , the tangen t plane to S 2 at a = ( a 1 , a 2 , a 3 , a 4 ) is giv en by a 1 x + a 2 y + a 3 z − a 4 w = 0 . Remark: If a ∈ S = V ( F ), as F ( a ) = P d +1 i =1 F x i ( a ) a i = 0 (b y Euler), the equ ation of the tangen t plane, T a S , t o S at a can also b e written as d +1 X i =1 F x i ( a )( x i − a i ) = 0 . 5.3. T ANGENT SP A CES OF HYPERSURF ACE S 85 No w, if a = ( a 1 : · · · : a d : 1) is a p oin t in the aﬃne patch U d +1 , then the equation of the in tersection of T a S with U d +1 is obtained b y setting a d +1 = x d +1 = 1, that is d X i =1 F x i ( a 1 , . . . , a d , 1)( x i − a i ) = 0 , whic h is just the equation of the aﬃne h yp erplane to S ∩ U d +1 at a ∈ U d +1 . It will b e con v enien t to adopt the follo wing notational con v en tion: G iv en an y p oin t, x = ( x 1 , . . . , x d ) ∈ R d , written as a ro w v ector, w e let x denote the correspo nding column v ector suc h that x ⊤ = x . Pro jectivities behav e w ell with resp ect to hy p ersurfaces and their tangen t spaces. Let S = V ( F ) ⊆ P d b e a pro jectiv e h yp ersurface, where F is a homogeneous p olynomial of degree m and let h : P d → P d b e a pro jectivit y (a bijectiv e pro jectiv e map). Assume that h is induced b y the in v ertible ( d + 1 ) × ( d + 1) matrix, A = ( a i j ), and write A − 1 = ( a − 1 i j ). F or an y h yp erplane, H ⊆ R d +1 , if ϕ is an y linear from deﬁning ϕ , i.e. , H = Ker ( ϕ ), then h ( H ) = { h ( x ) ∈ R d +1 | ϕ ( x ) = 0 } = { y ∈ R d +1 | ( ∃ x ∈ R d +1 )( y = h ( x ) , ϕ ( x ) = 0 ) } = { y ∈ R d +1 | ( ϕ ◦ h − 1 )( y ) = 0 } . Conseque ntly , if H is giv en b y α 1 x 1 + · · · + α d +1 x d +1 = 0 and if w e write α = ( α 1 , . . . , α d +1 ), then h ( H ) is the h yp erplane giv en b y the equation αA − 1 y = 0 . Similarly , h ( S ) = { h ( x ) ∈ R d +1 | F ( x ) = 0 } = { y ∈ R d +1 | ( ∃ x ∈ R d +1 )( y = h ( x ) , F ( x ) = 0) } = { y ∈ R d +1 | F (( A − 1 y ) ⊤ ) = 0 } is the h yp ersurface deﬁned b y the p olynomial G ( x 1 , . . . , x d +1 ) = F d +1 X j =1 a − 1 1 j x j , . . . , d +1 X j =1 a − 1 d +1 j x j ! . F urthermore, using the c hain rule, w e get ( G x 1 , . . . , G x d +1 ) = ( F x 1 , . . . , F x d +1 ) A − 1 , whic h sho ws that a p oin t, a ∈ S , is non-singular iﬀ its image, h ( a ) ∈ h ( S ), is non-singular on h ( S ). This also sho ws that h ( T a S ) = T h ( a ) h ( S ) , that is, the pro jectivit y , h , preserv es tangent spaces. In summary , we ha v e 86 CHAPTER 5. PROJECT IVE SP A CES AND POL YHEDRA, POLAR DUALITY Prop osition 5.6 L et S = V ( F ) ⊆ P d b e a pr oje ctive hyp ersurfac e, wher e F is a homo ge- ne o us p olynomial of d e gr e e m and let h : P d → P d b e a pr oje ctivity (a bije ctive pr oje ctive map). Then , h ( S ) is a hyp ersurfac e in P d and a p o i n t, a ∈ S , is nonsingular for S iﬀ h ( a ) is nonsingular for h ( S ) . F urthermor e, h ( T a S ) = T h ( a ) h ( S ) , that i s , the pr o je ctivity, h , pr eserves tangen t sp ac es. Remark: If h : P m → P n is a pro jectiv e map, say induced b y a n injectiv e linear map giv en b y the ( n + 1) × ( m + 1) matrix, A = ( a i j ), giv en any h yp ersurface, S = V ( F ) ⊆ P n , we can deﬁne the pul l-b ack , h ∗ ( S ) ⊆ P m , of S , b y h ∗ ( S ) = { x ∈ P m | F ( h ( x )) = 0 } . This is indeed a h ypersurface b ecause F ( x 1 , . . . , x n +1 ) is a homogenous p olynomial and h ∗ ( S ) is the zero lo cus o f the homogeneous p olynomial G ( x 1 , . . . , x m +1 ) = F m +1 X j =1 a 1 j x j , . . . , m +1 X j =1 a n +1 j x j ! . If m = n and h is a pro jectivit y , then w e ha v e h ( S ) = ( h − 1 ) ∗ ( S ) . 5.4 Quadrics (Aﬃne , Pro jectiv e) and P olar Du alit y The case where S = V ( Φ) ⊆ P d is a hy p ersurface giv en b y a homog eneous p olynomial, Φ( x 1 , . . . , x d +1 ), of degree 2 will come up a lot a nd deserv es a little more atten tion. In this case, if w e write x = ( x 1 , . . . , x d +1 ), then Φ( x ) = Φ( x 1 , . . . , x d +1 ) is completely determined b y a ( d + 1) × ( d + 1) symmetric matrix, say F = ( f i j ), and w e hav e Φ( x ) = x ⊤ F x = d +1 X i,j =1 f i j x i x j . Since F is symmetric, w e can write Φ( x ) = d +1 X i,j =1 f i j x i x j = d +1 X i =1 f i i x 2 i + 2 d +1 X i,j =1 i 0 for all i , 0 ≤ i ≤ n . Then, for ev ery x ∈ σ , t here is a unique face s suc h that x ∈ In t s , the face generated by those po in t s a i for whic h λ i > 0, where ( λ 0 , . . . , λ n ) are the barycen tric co ordinates of x . A simplex σ is conv ex, arcwise connected, compact, and closed. The in terior Int σ of a simplex is con v ex, arcwise connected, o p en, and σ is the closure of In t σ . W e no w put simplic es together to form more complex shap es, follo wing Munkre s [28]. The in tuition b ehind the next deﬁnition is that the building blo c ks should b e “glued cleanly ”. Deﬁnition 6.2 A simplici a l c omplex in E m (for short, a c omplex in E m ) is a set K consisting of a (ﬁnite or inﬁnite) set of simplices in E m satisfying the follo wing conditions: (1) Eve ry face of a simplex in K also belongs to K . (2) F or any t wo simplice s σ 1 and σ 2 in K , if σ 1 ∩ σ 2 6 = ∅ , then σ 1 ∩ σ 2 is a common face of b oth σ 1 and σ 2 . Ev ery k -simplex , σ ∈ K , is called a k -fac e (or fac e ) of K . A 0-face { v } is called a vertex and a 1- f ace is called an e dge . The dimension o f the simplicial comple x K is the maxim um of the dimensions of all simplic es in K . If dim K = d , then ev ery face of dimension d is called a c el l and ev ery face of dimension d − 1 is called a fac et . Condition (2) g uaran tees that the v arious simplices fo rmin g a complex inte rsect nicel y . It is easily sho wn that the follow ing condition is equiv alen t to condition (2): (2 ′ ) F or an y tw o distinct simplices σ 1 , σ 2 , In t σ 1 ∩ Int σ 2 = ∅ . Remarks: 1. A simplicial complex, K , is a com binatorial ob ject, namely , a set of simplices satisfyin g certain conditions but not a subs et of E m . Ho w ev er, ev ery complex, K , yields a subset of E m called the geometric realization of K a nd denoted | K | . This ob ject will b e deﬁned shortly and should not be confused with the comple x. Figure 6.1 illustrates this asp ect of the deﬁnition of a complex. F or clarit y , the t w o triangles (2 -simplic es) are drawn as disjoin t ob jects ev en though they share the common edge, ( v 2 , v 3 ) (a 1-simplex) and similarly for the edges that meet at some common v ertex. 6.1. SIMP LICIAL AND POL YHEDRAL COMPLEXES 97 v 1 v 2 v 3 v 3 v 2 v 4 Figure 6.1: A set of simplices forming a complex Figure 6.2: Collections of simplices not forming a complex 2. Some authors deﬁne a fac et o f a complex, K , of dimension d t o b e a d -simplex in K , as o pp osed to a ( d − 1)-simplex, as w e did. This practice is not consiste nt with the notion of facet of a p olyhedron and this is why w e prefer the terminology c el l f o r the d -simplices in K . 3. It is imp ortan t to note that in order for a complex, K , of dimens ion d to b e realized in E m , the dimension of the “ambien t space”, m , mus t b e big enough. F or example, there are 2-complex es that can’t b e realized in E 3 or ev en in E 4 . There has t o b e enough ro om in order for condition (2) to b e satisﬁed. It is not hard to pro v e that m = 2 d + 1 is alw ay s suﬃcie nt. Sometimes, 2 d w orks, f o r examp le in the case of surfaces (where d = 2). Some collections of simplices violating some of the conditions of Deﬁnition 6.2 are shown in F igure 6.2. On the left, the inte rsection of the t w o 2-simplices is neither an edge nor a v ertex of either tria ngle. In the middle case, tw o simplice s meet along an edge whic h is not an edge of either triangle. On the righ t, there is a missing edge and a missing v ertex. Some “legal” simplicial complexes are sho wn in Figure 6.4. The union | K | of all the simplices in K is a subset of E m . W e can deﬁne a to polog y on | K | b y deﬁning a subse t F of | K | to b e closed iﬀ F ∩ σ is closed in σ for ev ery face 98 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY v 1 v 2 v 3 v 4 Figure 6.3: The geometric realization of the complex of Figure 6.1 Figure 6.4: Examples of simplicial complexes σ ∈ K . It is immediately v eriﬁed that the axioms of a to p ological space are indeed satisﬁed. The resulting top ological space | K | is called the ge ometric r e alization of K . The geometric realization of the complex from Figure 6.1 is sho wn in Figure 6.3. Ob viously , | σ | = σ for ev ery simplex, σ . Also, note that distinct complex es ma y ha v e the same geometric realization. In fact, all the complexes obtained by sub divid ing the simplices of a giv en complex yield the same geometric realization. A p olytop e is the geometric r ealization of some simplic ial complex. A p olytop e of di- mension 1 is usually called a p olygon , and a p olytope of dimension 2 is usually called a p ol yhe dr on . When K consists of inﬁnitely man y simplices w e usually require that K b e lo c al ly ﬁnite , whic h means that ev ery v ertex b elongs to ﬁnitely man y faces . If K is lo cally ﬁnite, then its geometric realization, | K | , is lo cally compact. In t he sequel, w e will consider only ﬁnite simplicial complexes, that is, complexes K consisting of a ﬁnite n um b er of simplices. In this case, the top ology of | K | deﬁned ab o v e is iden tical to the topo logy induce d from E m . Also, for any simplex σ in K , In t σ coincides with the in terior ◦ σ of σ in the top ological sens e, and ∂ σ coincides with the bo undary of σ in the top ological sens e. 6.1. SIMP LICIAL AND POL YHEDRAL COMPLEXES 99 (a) (b) v Figure 6.5: (a) A complex that is not pure. (b) A pure complex Deﬁnition 6.3 Giv en an y complex, K 2 , a subset K 1 ⊆ K 2 of K 2 is a sub c omple x of K 2 iﬀ it is a lso a complex. F or an y complex, K , of dimension d , for an y i with 0 ≤ i ≤ d , the subset K ( i ) = { σ ∈ K | dim σ ≤ i } is called the i -skeleton of K . Clearly , K ( i ) is a subcomplex of K . W e also let K i = { σ ∈ K | dim σ = i } . Observ e that K 0 is the set of v ertices of K and K i is not a complex. A simplicial complex, K 1 is a sub division o f a complex K 2 iﬀ | K 1 | = | K 2 | and if eve ry face of K 1 is a subs et of some face of K 2 . A complex K of dimension d is pur e (or hom o gene ous ) iﬀ ev ery face of K is a face of some d -simplex of K (i.e., some cell of K ). A complex is c onne cte d iﬀ | K | is connected. It is easy to see that a complex is connected iﬀ its 1-sk eleton is connected. The in tuition b ehind the notion of a pure complex, K , of dimension d is that a pure complex is the result of gluing pieces all ha ving the same dimension, namely , d -simplices. F or example, in Figure 6.5, the complex on the left is not pure but the complex on the right is pure of dimension 2. Most of the shap es that w e will be in terested in are w ell appro ximated b y pure com- plexes, in particular, surfaces or solids. Ho w ev er, pure complexes ma y still ha v e undesirable “singularities” suc h as the verte x, v , in Figure 6.5(b). The notion of link of a v ertex prov ides a tec hnical w a y to deal with singularities. Deﬁnition 6.4 Let K b e any complex and let σ b e an y face o f K . The star , St( σ ) (or if w e need to b e v ery precise , St( σ , K )), o f σ is the sub complex of K consisting of all faces, τ , con taining σ and of all faces of τ , i.e. , St( σ ) = { s ∈ K | ( ∃ τ ∈ K )( σ  τ a nd s  τ ) } . 100 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY (a) v (b) v Figure 6.6: (a) A complex. (b) Star a nd Link of v The link , Lk( σ ) (or Lk( σ , K )) of σ is the subcomplex of K consisting of a ll faces in St( σ ) that do not in tersect σ , i.e., Lk( σ ) = { τ ∈ K | τ ∈ St( σ ) and σ ∩ τ = ∅} . T o simplify notation, if σ = { v } is a v ertex w e write St( v ) for St( { v } ) a nd Lk( v ) for Lk( { v } ). Figure 6.6 sho ws: (a) A comple x (on the left). (b) The star of the v ertex v , indicated in gray a nd t he link of v , show n as thick er lines. If K is pure and of dimension d , then St( σ ) is also pure of dimension d and if dim σ = k , then Lk( σ ) is pure of dimens ion d − k − 1. F o r tec hnical reasons , follo wing Munkres [2 8], besides deﬁning the comple x, St( σ ), it is useful to in tro duce t he op en star of σ , denoted st( σ ), deﬁne d as the subs pace of | K | consisting of the union of the in teriors, Int( τ ) = τ − ∂ τ , of all the f a ces , τ , con taining, σ . According to this deﬁnition, the op en star of σ is not a complex but instead a subset of | K | . Note that st( σ ) = | St( σ ) | , that is, the closure of st( σ ) is the geometric realization of the complex St( σ ). Then, lk( σ ) = | Lk( σ ) | is the union of the simplices in St( σ ) that are disjoin t fro m σ . If σ is a v ertex, v , w e hav e lk( v ) = st( v ) − st( v ) . Ho w ev er, b ew are that if σ is not a ve rtex, then lk( σ ) is prop erly con tained in st( σ ) − st( σ )! One of the nice prop erties of the op en star, st ( σ ), of σ is that it is op en. T o see this, observ e that for an y po in t, a ∈ | K | , there is a unique smallest simple x, σ = ( v 0 , . . . , v k ), suc h that a ∈ Int( σ ), that is, suc h that a = λ 0 v 0 + · · · + λ k v k 6.1. SIMP LICIAL AND POL YHEDRAL COMPLEXES 101 with λ i > 0 for all i , with 0 ≤ i ≤ k (and of course , λ 0 + · · · + λ k = 1). (When k = 0, w e ha v e v 0 = a and λ 0 = 1.) F o r ev ery a rbitrary vertex , v , o f K , w e deﬁne t v ( a ) b y t v ( a ) =  λ i if v = v i , with 0 ≤ i ≤ k , 0 if v / ∈ { v 0 , . . . , v k } . Using the ab o v e notation, observ e that st( v ) = { a ∈ | K | | t v ( a ) > 0 } and thus, | K | − st( v ) is the union of all the faces of K that do not contain v as a v ertex, ob viously a closed set. Th us, st( v ) is op en in | K | . It is also quite clear that st( v ) is path connected. Moreo v er, for a n y k -face, σ , of K , if σ = ( v 0 , . . . , v k ), then st( σ ) = { a ∈ | K | | t v i ( a ) > 0 , 0 ≤ i ≤ k } , that is, st( σ ) = st ( v 0 ) ∩ · · · ∩ st( v k ) . Conseque ntly , st( σ ) is op en and path connected.  Unfortunately , the “nice” equation St( σ ) = St( v 0 ) ∩ · · · ∩ St( v k ) is false! (and anagolously for Lk( σ ).) F or a coun ter-example, consider the b oundary of a tetrahedron with one face remo v ed. Recall that in E d , the (op en) unit b a l l, B d , is deﬁned b y B d = { x ∈ E d | k x k < 1 } , the close d unit b al l, B d , is deﬁned b y B d = { x ∈ E d | k x k ≤ 1 } , and the ( d − 1) -spher e , S d − 1 , b y S d − 1 = { x ∈ E d | k x k = 1 } . Ob viously , S d − 1 is the b oundary of B d (and B d ). Deﬁnition 6.5 Let K b e a pure complex of dimension d and let σ b e any k -fa ce of K , with 0 ≤ k ≤ d − 1. W e sa y that σ is nonsingular iﬀ the geometric realization, lk( σ ), of the link of σ is homeomorphic to either S d − k − 1 or to B d − k − 1 ; this is written as lk( σ ) ≈ S d − k − 1 or lk( σ ) ≈ B d − k − 1 , where ≈ means homeomorphic. 102 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY In Figure 6.6, note that the link of v is not homeomorphic to S 1 or B 1 , so v is singular. It will also b e useful to expre ss St( v ) in terms of Lk( v ), where v is a v ertex, and for this, w e deﬁne y et another notio n of cone. Deﬁnition 6.6 Giv en an y complex, K , in E n , if dim K = d < n , for an y p oin t, v ∈ E n , suc h that v do es not belong to the aﬃne h ull of | K | , the c o n e on K with vertex v , denoted, v ∗ K , is the complex consisting of all simplices of the form ( v , a 0 , . . . , a k ) and their faces, where ( a 0 , . . . , a k ) is an y k -face of K . If K = ∅ , we set v ∗ K = v . It is not hard to c hec k that v ∗ K is indeed a complex of dimension d + 1 con taining K as a subcomplex. Remark: Unfortunately , the w ord “cone” is ov erloaded. It migh t ha v e b een b etter to use the lo cution pyr amid instead of cone as some authors do (for example, Ziegler). Ho w ev er, since w e ha v e b een follo wing Munkres [28], a standard referenc e in algebraic top ology , w e decided to stic k with the terminology used in that b o ok, namely , “cone”. The follo wing prop osition is also easy to pro v e: Prop osition 6.1 F or any c om p lex, K , of dimension d and any ve rtex, v ∈ K , we have St( v ) = v ∗ Lk( v ) . Mor e gener al ly, for any fac e, σ , of K , we hav e st( σ ) = | St( σ ) | ≈ σ × | v ∗ Lk( σ ) | , for every v ∈ σ and st( σ ) − st( σ ) = ∂ σ × | v ∗ Lk ( σ ) | , for every v ∈ ∂ σ . Figure 6.7 sho ws a 3-dimensional complex. The link of the edge ( v 6 , v 7 ) is the pentagon P = ( v 1 , v 2 , v 3 , v 4 , v 5 ) ≈ S 1 . The link of the verte x v 7 is the cone v 6 ∗ P ≈ B 2 . The link of ( v 1 , v 2 ) is ( v 6 , v 7 ) ≈ B 1 and the link o f v 1 is the union of the triangles ( v 2 , v 6 , v 7 ) and ( v 5 , v 6 , v 7 ), whic h is homeomorphic to B 2 . Giv en a pure complex, it is necessary to distinguish b et w een tw o kinds of faces. Deﬁnition 6.7 Let K b e an y pure complex of dimension d . A k -face, σ , of K is a b oundary or ex terna l face iﬀ it belongs to a single cell (i.e., a d -simplex) of K a nd otherwise it is called an internal face (0 ≤ k ≤ d − 1). The b oundary of K , denoted b d( K ), is the subcomplex of K consisting of all b oundary facets of K together with their faces. 6.1. SIMP LICIAL AND POL YHEDRAL COMPLEXES 103 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Figure 6.7: More examples of links and stars It is clear b y deﬁnition that b d( K ) is a pure complex of dimens ion d − 1. Ev en if K is connected, b d( K ) is not connec ted, in general. F o r examp le, if K is a 2-complex in the plane, the b oundary of K usually consists of sev eral simple closed p olygons (i.e, 1 dimensional complexes homeomorphic to the circle, S 1 ). Prop osition 6.2 L et K b e any pur e c omplex of dime nsion d . F or any k -fac e, σ , of K the b ounda ry c omplex, b d(Lk( σ )) , is non e mpty iﬀ σ is a b oundary fac e of K ( 0 ≤ k ≤ d − 2 ). F urthermor e, Lk bd ( K ) ( σ ) = b d(Lk ( σ )) for every fac e , σ , of b d( K ) , wher e Lk bd ( K ) ( σ ) denotes the link of σ in b d( K ) . Pr o of . Let F b e an y facet o f K con taining σ . W e ma y assume that F = ( v 0 , . . . , v d − 1 ) and σ = ( v 0 , . . . , v k ), in whic h case, F ′ = ( v k +1 , . . . , v d − 1 ) is a ( d − k − 2)-face of K and b y deﬁnition of Lk( σ ), we ha v e F ′ ∈ Lk( σ ). Now , ev ery cell (i.e., d -simplex), s , con taining F is of the form s = con v ( F ∪ { v } ) for some v ertex, v , and s ′ = con v ( F ′ ∪ { v } ) is a ( d − k − 1)- face in Lk ( σ ) con taining F ′ . Consequen tly , F ′ is an external face of Lk( σ ) iﬀ F is an exte rnal facet of K , establishing the prop osition. The second statemen t follo ws immediately fro m the pro of of the ﬁrst. Prop osition 6.2 shows that if ev ery f a ce of K is nonsingular, then the link of eve ry in ternal face is a sphere whereas the link of ev ery external face is a ball. The follo wing prop osition sho ws that for any pure complex, K , nonsingularit y of all the v ertices is enough to imply that ev ery op en star is homeomorphic to B d : Prop osition 6.3 L et K b e any pur e c o m plex of dimension d . If every vertex o f K is non- singular, then st( σ ) ≈ B d for eve ry k -fac e, σ , of K ( 1 ≤ k ≤ d − 1 ). 104 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY Pr o of . Let σ be an y k -face of K and assum e that σ is generated b y the v ertices v 0 , . . . , v k , with 1 ≤ k ≤ d − 1. By h ypo thes is, lk( v i ) is homeomorphic to either S d − 1 or B d − 1 . Then, it is easy to sho w that in either case, w e hav e | v i ∗ Lk( v i ) | ≈ B d , and b y Prop osition 6.1, w e get | St( v i ) | ≈ B d . Conseque ntly , st( v i ) ≈ B d . F urthermore, st( σ ) = st( v 0 ) ∩ · · · ∩ st( v k ) ≈ B d and so, st( σ ) ≈ B d , as claimed. Here are more useful prop ositions ab out pure complexes without singularities. Prop osition 6.4 L et K b e any pur e c o m plex of dimension d . If every vertex o f K is non- singular, then for every p oi n t, a ∈ | K | , ther e is an op en subset, U ⊆ | K | , c ontaining a such that U ≈ B d or U ≈ B d ∩ H d , whe r e H d = { ( x 1 , . . . , x d ) ∈ R d | x d ≥ 0 } . Pr o of . W e already know from Prop osition 6.3 that st( σ ) ≈ B d , for ev ery σ ∈ K . So, if a ∈ σ and σ is not a b oundary face, we can tak e U = st( σ ) ≈ B d . If σ is a b oundary fa ce, then | σ | ⊆ | b d(St( σ )) | and it can b e sho wn that we can tak e U = B d ∩ H d . Prop osition 6.5 L et K b e any pur e c omplex of dimension d . If every fac et of K is nonsin- gular, then every fac et of K , is c ontaine d in at most two c el ls ( d -simplic es). Pr o of . If | K | ⊆ E d , then this is an immediate conseq uence of the deﬁnition of a complex. Otherwise, consider lk( σ ). By h yp othesis, either lk( σ ) ≈ B 0 or lk( σ ) ≈ S 0 . As B 0 = { 0 } , S 0 = {− 1 , 1 } and dim Lk( σ ) = 0, w e deduce that Lk( σ ) has either one or t wo p oints , whic h pro v es that σ b elongs to at most tw o d - simp lices. Prop osition 6.6 L et K b e any pur e and c onne cte d c omplex of dimen s ion d . If every f a c e of K is nonsingular, then for every p air of c el ls ( d -s implic es), σ and σ ′ , ther e is a se quenc e of c el ls, σ 0 , . . . , σ p , with σ 0 = σ an d σ p = σ ′ , and s uch that σ i and σ i +1 have a c ommon fac et, for i = 0 , . . . , p − 1 . Pr o of . W e pro cee d b y induction on d , using the fact t hat the links are connected f o r d ≥ 2 . Prop osition 6.7 L et K b e any pur e c omplex of dimension d . If every fac et of K is nonsin- gular, then the b oundary, b d( K ) , of K is a pur e c om plex of dimension d − 1 with an empty b ounda ry. F urthermor e, if every fac e of K is nonsingular, then every fac e of b d( K ) is also nonsingular. 6.1. SIMP LICIAL AND POL YHEDRAL COMPLEXES 105 Pr o of . Left as an exercise. The building blocks of simplicial complexes, namely , simplicie s, a re in some sense math- ematically ideal. Ho w ev er, in practice, it may b e desirable to use a more ﬂexible set of building blo c ks. W e can indee d do this and use con v ex p olytop es a s o ur building blo c ks. Deﬁnition 6.8 A p olyhe dr al c omplex in E m (for short, a c omplex in E m ) is a set, K , consist- ing of a (ﬁnite or inﬁnite) set of con v ex p olytop es in E m satisfying the follo wing conditions: (1) Eve ry face of a p olytope in K also b elongs to K . (2) F or any tw o p olytop es σ 1 and σ 2 in K , if σ 1 ∩ σ 2 6 = ∅ , then σ 1 ∩ σ 2 is a common f ace of b oth σ 1 and σ 2 . Ev ery p olytope, σ ∈ K , o f dimension k , is called a k -fac e (or fac e ) of K . A 0- face { v } is called a vertex and a 1-fa ce is called an e dge . The dimensi o n of the p olyhedral complex K is the maxim um of the dimensions of all p olytop es in K . If dim K = d , then ev ery face of dimension d is called a c el l and ev ery face of dimension d − 1 is called a fac et . Remark: Since the building blo c ks o f a p olyhedral complex are conv ex p olytop es it migh t b e more appropriate to use the term “p olytopal complex” rather than “p olyhedral complex” and some authors do that. On the other hand, most of the traditional litterature uses the terminology p olyhe dr al c omplex so w e will stic k to it. There is a notion o f complex where the building blo c ks are cones but these are called fans . Ev ery conv ex p olytope, P , yields t w o natural p olyhedral complexes: (i) The p olyhedral complex, K ( P ), consisting of P together with all of its faces. This complex has a single cell, namely , P itself. (ii) The b o und a ry c omplex , K ( ∂ P ), consisting of all faces of P other than P itself. The cells of K ( ∂ P ) are the facets of P . The notions of k -sk eleton a nd purene ss are deﬁned just as in the simplicial case. The notions of star and link ar e deﬁned for p olyhedral complex es just as they are deﬁned f or simplicial complex es exc ept that the word “ f ace” now means face of a p olytop e. Now , by Theorem 4.7, ev ery p olytope, σ , is the conv ex h ull of its v ertices. Let v ert( σ ) denote the set of v ertices of σ . Then, w e hav e the follo wing crucial observ ation: Giv en a ny p olyhedral complex, K , for ev ery p oin t, x ∈ | K | , there is a unique p olytop e, σ x ∈ K , suc h that x ∈ In t( σ x ) = σ x − ∂ σ x . W e deﬁne a function, t : V → R + , that tests whe ther x b elongs to the in terior of an y face (p olytop e) of K having v as a v ertex as fo llo ws: F or ev ery vertex , v , of K , t v ( x ) =  1 if v ∈ v ert( σ x ) 0 if v / ∈ v ert( σ x ), where σ x is the unique face of K suc h that x ∈ In t( σ x ). 106 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY No w, just as in the simplicial case, the op en star, st( v ), of a v ertex, v ∈ K , is giv en b y st( v ) = { x ∈ | K | | t v ( x ) = 1 } and it is an op en subset of | K | (t he set | K | − st( v ) is the union of the p olytop es of K that do not con tain v as a v ertex, a closed subset of | K | ). Also, for any fa ce, σ , of K , the op en star, st( σ ), of σ is giv en b y st( σ ) = { x ∈ | K | | t v ( x ) = 1 , for all v ∈ v ert( σ ) } = \ v ∈ vert( σ ) st( v ) . Therefore, st( σ ) is also op en in | K | . The next prop osition is another result that seems quite ob vious, y et a rigorous pro of is more in v olv ed that w e migh t think. This prop osition states that a con v ex p olytop e can alw a ys b e cut up in to simplice s, that is, it can b e sub divided in to a simplicial complex. In o ther w ords, ev ery con v ex p olytop e can b e triangulated. Th is implies that simplicial complexes are as general as p olyhed ral comple xes. One should b e warne d that ev en though, in the plane, ev ery b o und ed region (not nec- essarily con v ex) whose b o undary consists of a ﬁnite num b er of closed p olygons (po lygons homeomorphic to the circle, S 1 ) can b e triangulated, this is no longer true in three dimen- sions! Prop osition 6.8 Every c onvex d -p olytop e , P , c an b e sub divide d into a simplicial c omplex without adding any new vertic es, i.e., every c onvex p olytop e c an b e triangulate d. Pr o of ske tch . It w ould b e tempting to pro ceed by induction on the dimens ion, d , o f P but w e do not kno w a n y correct pro of of this kind. Instead, we pro ceed by induction o n the n um b er, p , of v ertices of P . Since dim( P ) = d , w e mu st ha v e p ≥ d + 1. The case p = d + 1 correspo nds to a simplex, so the base case holds. F o r p > d + 1, w e can pic k some v ertex, v ∈ P , such that the con v ex h ull, Q , of the remaining p − 1 v ertices still has dimension d . Then, b y the induction h yp othesi s, Q , has a simp licial subdivision. Now , w e sa y that a facet, F , of Q is visible fr om v iﬀ v and the in terior of Q are strictly separated b y the supp orting h yp erplane of F . Then, w e add the d -simplices, con v( F ∪ { v } ) = v ∗ F , for ev ery f a cet, F , of Q visible from v to those in the triangulation of Q . W e claim that the resulting collection of simplices (with their faces) constitutes a simplicial complex sub dividing P . This is the part of the pro of that requires a careful a nd somewhat tedious case analysis, whic h w e o mit. Ho w ev er, the reader should c hec k that ev erything really w orks out! With all this preparation, it is no w quite natural to deﬁne com binatorial manifolds. 6.2. COMBINA TORIAL AND TOPOLOGICAL MANIF OLDS 107 6.2 Com binatorial and T o p ological Manifold s The notion of pure complex without singular faces turns out to b e a v ery go o d “discre te” appro ximation of the notion of (top ological) manifold b ecause of its highly computational nature. This motiv ates the follo wing deﬁnition: Deﬁnition 6.9 A c ombinatorial d -man i fold is any space, X , homeomorphic to the geometric realization, | K | ⊆ E n , o f some pure (simplicial or p olyhedral) complex, K , of dimension d whose faces are all nonsingular. If the link of ev ery k -face of K is homeomorphic to the sphere S d − k − 1 , we sa y that X is a comb inatorial manifold without b o und a ry , else it is a com binatorial manifold with b ound a ry . Other authors use the term triangulation for what w e call a comb inatorial manifold. It is easy to see that the connected componen ts of a com binatorial 1-manifold are either simple closed p olygons or simple c hains (“simple” means that the inte riors of distinct edges are disjoin t). A com binatorial 2-manifold whic h is conne cted is a lso called a c ombinatorial surfac e (with o r without b oundary). Prop osition 6.7 immediately yields the f ollo wing result: Prop osition 6.9 If X is a c ombinatorial d -manifold with b oundary, then b d( X ) is a c om- binatorial ( d − 1 )-man i f o ld without b oundary. No w, b ecause w e are assuming that X sits in some Euclidean space, E n , the space X is Hausdorﬀ and sec ond-countable . (Recall that a topolo g ic al space is second-coun t a ble iﬀ there is a countable family , { U i } i ≥ 0 , of o p en sets of X suc h that eve ry op en subset of X is the union of o pen sets from this family .) Since it is desirable to hav e a go o d matc h b et w een manifolds and com binatorial manifolds, w e are led to the deﬁnition b elo w. Recall that H d = { ( x 1 , . . . , x d ) ∈ R d | x d ≥ 0 } . Deﬁnition 6.10 F o r an y d ≥ 1 , a (top ol o gic al) d -manifold with b oundary is a second- coun table, top ological Hausdorﬀ space M , together with an op en co v er, ( U i ) i ∈ I , o f op en sets in M and a family , ( ϕ i ) i ∈ I , of homeomorphisms , ϕ i : U i → Ω i , where eac h Ω i is some op en subse t of H d in the subset top ology . Eac h pair ( U, ϕ ) is called a c o or dinate system , or chart , of M , each homeomorphis m ϕ i : U i → Ω i is called a c o or dinate map , and its in vers e ϕ − 1 i : Ω i → U i is called a p ar ame teriza tion of U i . The family ( U i , ϕ i ) i ∈ I is often called an atlas for M . A (top olo gic al) b or der e d surfac e is a connected 2-manifold with b oundary . If for ev ery homeomorphism, ϕ i : U i → Ω i , the op en set Ω i ⊆ H d is actually an op en set in R d (whic h means that x d > 0 for ev ery ( x 1 , . . . , x d ) ∈ Ω i ), then w e say that M is a d -ma nifold . Note that a d -manifold is also a d -manifold with b oundary . If ϕ i : U i → Ω i is some homeomorphism onto some op en set Ω i of H d in the subset top ology , some p ∈ U i ma y b e mapp ed in to R d − 1 × R + , or in to the “b oundary” R d − 1 × { 0 } 108 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY of H d . L etting ∂ H d = R d − 1 × { 0 } , it can b e sho wn using homology that if some co ordinate map, ϕ , deﬁned on p maps p into ∂ H d , then ev ery co ordinate map, ψ , deﬁned on p maps p in to ∂ H d . Th us, M is the disjoin t union of tw o sets ∂ M and In t M , where ∂ M is the subset consisting of all p o in t s p ∈ M that are mapp ed b y some (in fact, all) co ordinate map, ϕ , deﬁned on p in to ∂ H d , and where Int M = M − ∂ M . The set ∂ M is called the b oundary of M , and the set In t M is called the interior of M , ev en though t his terminology clashes with some prior top ological deﬁnitions. A go o d example of a b ordered surface is the M¨ obius strip. The b oundary of the M¨ o bius strip is a circle. The b oundary ∂ M of M may b e empt y , but Int M is nonempt y . Also, it can be show n using homology that the in teger d is unique. It is clear that In t M is op en and a d - manifold, and that ∂ M is closed. If p ∈ ∂ M , and ϕ is some co ordinate map deﬁned on p , since Ω = ϕ ( U ) is a n op en subset of ∂ H d , there is some o p en half ball B d o + cen tered at ϕ ( p ) and con tained in Ω whic h in tersects ∂ H d along an op en ba ll B d − 1 o , and if w e consider W = ϕ − 1 ( B d o + ), w e hav e an op en subset of M con taining p whic h is mapp ed homeomorphically o n t o B d o + in suc h that w a y that ev ery p oin t in W ∩ ∂ M is mapp ed onto the op en ball B d − 1 o . Th us, it is easy to see that ∂ M is a ( d − 1) - manifold. Prop osition 6.10 Every c ombinatorial d -manifold is a d -manifold with b oundary. Pr o of . This is a n immed iate conseq uence of Prop osition 6.4. Is the con v erse of Proposition 6.10 true? It turns out that answ er is y es for d = 1 , 2 , 3 but no for d ≥ 4. This is not hard to pro v e for d = 1. F or d = 2 and d = 3, this is quite hard to prov e; among other things, it is necess ary to pro v e that triangulations exis t and this is v ery tec hnical. F or d ≥ 4, not ev ery manifold can b e triang ulated (in f act, this is undecidable!). What if w e assume that M is a triang ulated manifold, whic h means t hat M ≈ | K | , for some pure d -dimensional complex, K ? Surprinsingly , for d ≥ 5, there are triangulated manifolds whose links are not spherical (i.e., not homeomorphic to B d − k − 1 or S d − k − 1 ), see Th urston [39]. F o r tunately , w e will only hav e to deal with d = 2 , 3! Another issue that mus t b e a ddres sed is orien tabilit y . Assume that w e ﬁx a total ordering of the v ertices of a complex, K . Let σ = ( v 0 , . . . , v k ) b e any simplex. Recall that eve ry permutation (of { 0 , . . . , k } ) is a pro duct of tr a n sp ositions , where a transp osition sw aps tw o distinct elemen ts, say i and j , and lea v es ev ery other elemen t ﬁxed. F urthermore, for an y p erm utation, π , the parit y of the n um b er o f transp ositions needed to obtain π only dep ends on π and it called the signatur e of π . W e say that two p erm utations ar e e q uiva lent iﬀ they ha v e the same signature. Consequen tly , there are t w o equiv alence classes of p erm utations: Those of ev en signature and those of o dd signature. 6.2. COMBINA TORIAL AND TOPOLOGICAL MANIF OLDS 109 Then, an orie ntation of σ is the c hoice of o ne o f the t w o equiv a lenc e classes of p erm utations of its v ertices. If σ has b een giv en an orientation, then w e denote b y − σ the result of assigning the other orien tation to it (w e call it the opp osite orientation ). F o r example, (0 , 1 , 2 ) has the t w o orien tation classes : { (0 , 1 , 2) , (1 , 2 , 0) , (2 , 0 , 1) } and { (2 , 1 , 0) , (1 , 0 , 2) , (0 , 2 , 1) } . Deﬁnition 6.11 Let X ≈ | K | b e a com binatorial d -manifold. W e sa y that X is orientable if it is p ossible t o assign an orien tation to all of its cells ( d -simplic es) so that whenev er t w o cells σ 1 and σ 2 ha v e a common facet, σ , the t wo or ientations induced b y σ 1 and σ 2 on σ are opp osite. A com binatorial d -manifold together with a sp eciﬁc orien tation of its cells is called an oriente d manifold . If X is not orien table w e sa y that it is non-orientable . Remark: It is p ossible to deﬁne the notion of orien tation of a manifold but this is quite tec hnical and w e prefer to a v oid digressing into this matter. This sho ws another adv an tage of com binatorial manifolds: The deﬁnition of o r ientabilit y is simple a nd quite natural. There are non-orientable (com binatorial) surfaces , for example, the M¨ obius strip whic h can b e realized in E 3 . The M¨ o bius strip is a surface with boundary , its b oundary b eing a circle. There are also non-orientable (com binatorial) surfaces suc h as the Klein b ottle or the pro jectiv e plane but they can only be realized in E 4 (in E 3 , they must ha ve singularities suc h as self-in tersection). W e will only b e dealing with orien table manifolds and, most of the time, surfaces. One o f the most imp ortan t inv arian ts o f com binatorial ( a nd top o logical) manifolds is their Euler(-Poinc ar´ e) char acteristic . In the next c hapter, w e pro ve a famous form ula due to Poincar ´ e giving the Euler c haracteristic of a con v ex p olytope. F or this, w e will intro duce a tec hnique of indep ende nt in terest called s h el ling . 110 CHAPTER 6. BASICS OF COMBINA TORIAL TOPOLOGY Chapter 7 Shellings, the Euler-P oi ncar ´ e F orm ula for P oly top es, the Dehn-Sommerville Equations and the Upp er Bound Theorem 7.1 Shelling s The notion of shellabilit y is motiv ated by the desire to giv e an inductiv e pro of of the Euler- P oincar ´ e formu la in an y dimension. Historically , this f o rm ula w as discov ered b y Euler fo r three dimensional p olytop es in 1752 (but it w as a lready kno wn to Descartes around 1640). If f 0 , f 1 and f 2 denote the nu mber of v ertices, edges and triangles of the three dimensional p olytope, P , (i.e., the num b er of i -faces of P fo r i = 0 , 1 , 2) , then the Euler formula states that f 0 − f 1 + f 2 = 2 . The pro of of Euler’s formula is not ve ry diﬃcult but one still has to exercise caution. Euler’s form ula w as generalized to arbitrary d -dimensional p olytop es b y Sch l¨ aﬂi (18 52) but the ﬁrst correct pro of w as giv en by P oincar ´ e. F or this, P oincar ´ e had to la y the f o undations of algebraic top ology and after a ﬁrst “pro of ” g iven in 18 9 3 (con taining some ﬂa ws) he ﬁnally ga v e the ﬁrst correct proo f in 1899. If f i denotes the nu mber of i -faces of the d - dimen sional p olytope, P , (with f − 1 = 1 and f d = 1), the Euler-Poinc ar´ e f o rmula states that: d − 1 X i =0 ( − 1) i f i = 1 − ( − 1) d , whic h can also b e written as d X i =0 ( − 1) i f i = 1 , 111 112 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA b y incorp orating f d = 1 in the ﬁrst formula or as d X i = − 1 ( − 1) i f i = 0 , b y incorp orating b oth f − 1 = 1 and f d = 1 in the ﬁrst formula. Earlier inductiv e “pro ofs” of the ab o v e form ula w ere prop ose d, notably a pro of by Sc hl¨ aﬂi in 1852, but it was later observ ed that all these pro ofs assume that the b oundary of ev ery p olytope can b e built up inductiv ely in a nice w ay , what is called shel lability . Actually , coun ter-examples of shellabilit y for v arious simplicial complexes suggested that po lytop es w ere perhaps not shellable. How ev er, the fact that p olytopes a r e shellable w as ﬁnally prov ed in 1970 b y Bruggesse r and Mani [12] and so on af t er that ( a lso in 1970) a striking application of shellabilit y w as made b y McMullen [27] who gav e the ﬁrst pro of of the so-called “upp er b ound theorem”. As shellabilit y of p olytop es is a n imp ortan t to ol and as it yields one of the cleanest inductiv e pro of of the Euler-P o incar ´ e for mula, w e will sk etc h its pro of in some details. This Chapter is heav ily inspired by Ziegler’s excellen t treatmen t [43], Chapter 8. W e b egin with the deﬁnition of shellabilit y . It’s a bit tec hnical, so please b e patien t! Deﬁnition 7.1 Let K b e a pure p olyhedral complex of dimens ion d . A shel ling of K is a list, F 1 , . . . , F s , o f the cells (i.e., d -faces) of K suc h that either d = 0 (and th us, all F i are p oin ts) or the follo wing conditions hold: (i) The b oundary complex, K ( ∂ F 1 ), of the ﬁrst cell, F 1 , of K has a shelling. (ii) F or an y j , 1 < j ≤ s , the in tersection of the cell F j with the previous cells is nonempt y and is an initial segmen t of a shelling of the ( d − 1)-dimensional b oundary complex of F j , that is F j ∩ j − 1 [ i =1 F i ! = G 1 ∪ G 2 ∪ · · · ∪ G r , for some shelling G 1 , G 2 , . . . , G r , . . . , G t of K ( ∂ F j ), with 1 ≤ r ≤ t . As the in tersection should b e the initial segmen t of a shelling for the ( d − 1)-dimensional complex, ∂ F j , it has to b e pure ( d − 1)- dimen sional and connected for d > 1. A p olyhedral complex is sh e l lable if it is pure and has a shelling. Note that shellabiliy is o nly deﬁned fo r pure complexes . Here are some examples of shellable complexe s: (1) Eve ry 0-dimensional complex, that is, eve y set of p oin ts, is shellable, b y deﬁnition. 7.1. SHELLINGS 113 1 2 1 2 3 4 5 6 7 8 1 2 3 4 5 Figure 7.1: Non shellable and Shellable 2-complexes (2) A 1-dimensional complex is a g ra ph without lo ops and par allel edges. A 1-dimensional complex is shellable iﬀ it is connected, whic h implies that it has no isolated vertic es. An y ordering o f the edges, e 1 , . . . , e s , such that { e 1 , . . . , e i } induces a connected sub- graph for ev ery i will do. Such an ordering can b e deﬁned inductiv ely , due to the connectivit y of the graph. (3) Eve ry simplex is shell able. In fact, an y ordering of its facets yield s a shelling. This is easily sho wn b y induction on the dimension, since the in tersection of any tw o facets F i and F j is a facet of b oth F i and F j . (4) The d -cub es are shellable. By induction on the dimension, it can b e sho wn that ev ery ordering of the 2 d f a cets F 1 , . . . , F 2 d suc h that F 1 and F 2 d are opp osite (that is, F 2 d = − F 1 ) yields a shelling. Ho w ev er, already for 2-complexe s, problems arise. F or examp le, in Figure 7.1, the left and the middle 2-complexes are not shellable but the righ t complex is shellable. The problem with the left complex is that cells 1 and 2 in tersect at a v ertex, whic h is not 1-dimensional, and in the middle complex, the in tersecti on of cell 8 with its predecessors is not connected. In con trast, the ordering of the right complex is a shelling. How ev er, observ e that the rev erse ordering is not a shelling b ecause cell 4 has an empt y inters ection with cell 5! Remarks: 1. Condition (i) in D eﬁni tion 7.1 is redundan t b ecaus e, as w e shall prov e shortly , ev ery p olytope is shellable. Ho w ev er, if w e w an t to use this deﬁnition for more g ene ral complexes , then condition (i) is necessary . 2. When K is a simplicial complex, condition (i) is of course redundan t, as ev ery simpl ex is shellable but condition (ii) can also b e simpliﬁed to: (ii’) F or any j , with 1 < j ≤ s , the in tersection of F j with the previous cells is nonempt y and pure ( d − 1)-dimensional. This means that for ev ery i < j there is some l < j suc h that F i ∩ F j ⊆ F l ∩ F j and F l ∩ F j is a facet of F j . 114 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA The follo wing prop osition yields an imp o rtan t piece of information ab out the lo cal struc- ture of shellable simplicial complexe s: Prop osition 7.1 L et K b e a shel lable sim plicial c omplex and say F 1 , . . . , F s is a she l ling for K . Then, for every vertex, v , the r estriction of the ab ove se quenc e to the link , Lk( v ) , and to the star, St( v ) , a r e sh e l lings. Since the complex, K ( P ), asso ciated with a p olytop e, P , has a single cell, namely P itself, note that by condition (i) in the deﬁnition of a shelli ng, K ( P ) is shell able iﬀ the complex, K ( ∂ P ), is shellable. W e will sa y simply sa y that “ P is shellable” instead of “ K ( ∂ P ) is shellable”. W e ha v e t he follo wing useful prop erty of shellings of p olytop es whose pro of is left as an exercise (use induction on the dimens ion): Prop osition 7.2 Given any p olytop e, P , i f F 1 , . . . , F s is a shel li n g of P , then the r everse se q uenc e F s , . . . , F 1 is also a shel li n g of P .  Prop osition 7.2 generally fails for complexes that are not p olytop es, see the righ t 2- complex in Figure 7.1. W e will no w presen t the pro of that ev ery p olytope is shellable, using a tec hnique in v en ted b y Bruggesser a nd Mani (1970) kno wn as line shel ling [12]. This is quite a simple and natural idea if one is willing to ignore the tec hnical details inv olved in actually c hec king that it w orks. W e b egin by explainin g this idea in the 2- dime nsional case, a con v ex p olygon, since it is particularly simple. Consider the 2 - polytop e, P , sho wn in Figure 7.2 (a p olygon) whose faces are lab eled F 1 , F 2 , F 3 , F 4 , F 5 . Pic k an y line, ℓ , inters ecting the in terior of P and in tersecting the sup- p orting lines of the facets of P ( i.e . , the edges of P ) in distinct p oin ts lab eled z 1 , z 2 , z 3 , z 4 , z 5 (suc h a line can alw ays b e found, a s will b e sho wn shortly). Orien t the line, ℓ , (sa y , up w ard) and tra v el on ℓ starting from the po in t of P where ℓ lea v es P , namely , z 1 . F or a while, only face F 1 is visible but when w e reac h the interse ction, z 2 , of ℓ with the supporting line of F 2 , the face F 2 b ecomes visible and F 1 b ecomes in visible a s it is now hidden by the supp orting line of F 2 . So far, w e ha v e seen the faces, F 1 and F 2 , in that or d e r . As we con tin ue trav eling along ℓ , no new face becomes visible but fo r a more complicated p olygon, other faces, F i , w ould b ecome visible o ne at a time as w e reach the in tersection, z i , of ℓ with the suppor t ing line of F i and the order in whic h these faces b ecome visible corresp onds to the ordering of the z i ’s along the line ℓ . Then, w e imagine that w e tra ve l v ery fast and when we reac h “+ ∞ ” in the up w ard direction on ℓ , w e instantly come bac k on ℓ fro m b elo w at “ −∞ ”. At this p oin t, w e only see the face of P correspo nding to the lo w est supp orting line of faces of P , i.e., the line corresponding to the smalles t z i , in our case, z 3 . A t this stage, the only visible face is F 3 . W e contin ue tra v eling up w ard on ℓ and w e reac h z 3 , the interse ction of the supp orting line of F 3 with ℓ . At this momen t, F 4 b ecomes visible and F 3 disappears as it is no w hidden 7.1. SHELLINGS 115 b y the supporting line of F 4 . Note that F 5 is not visib le at this stage. Finally , w e reach z 4 , the in tersection of the supp orting line of F 4 with ℓ and at this momen t, the last facet, F 5 , b ecomes visib le (and F 4 b ecomes in visible , F 3 b eing also invi sible). Our trip stops when w e reac h z 5 , the inte rsection of F 5 and ℓ . During the second phase of our trip, we sa w F 3 , F 4 and F 5 and the en tire trip yields the sequence F 1 , F 2 , F 3 , F 4 , F 5 , whic h is easily seen to b e a shelling of P . F 1 F 2 F 3 F 5 F 4 z 1 z 2 z 3 z 4 z 5 ℓ Figure 7.2: Shelling a p olygon b y tra v elling along a line This is the crux of the Bruggesser-Mani metho d for shellin g a p olytop e: W e tra v el alo ng a suitably c hosen line and record the order in whic h the faces b ecome visible during this trip. This is wh y suc h shellings are called line shel lings . 116 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA In order to prov e that p olytop es are shellable w e need the notion of p oin ts and lines in “ g ene ral p osition”. Recall from the equiv alence of V -p olytop es and H -p olytopes that a p olytope, P , in E d with nonempt y in terior is cut out b y t irredundan t h yp erplanes , H i , and b y pic king the origin in the in terior of P the equations of the H i ma y b e assumed to b e of the form a i · z = 1 where a i and a j are not prop ortional for all i 6 = j , so that P = { z ∈ E d | a i · z ≤ 1 , 1 ≤ i ≤ t } . Deﬁnition 7.2 Let P b e an y p olytope in E d with nonempt y in terior and assume that P is cut out b y the irredudan t h yp erplanes, H i , of equations a i · z = 1, for i = 1 , . . . , t . A po int, c ∈ E d , is said to b e in gener al p osi tion w.r.t. P is c do es not b elong to any of the H i , that is, if a i · c 6 = 1 for i = 1 , . . . , t . A line, ℓ , is said to b e in ge n er al p osition w.r.t. P if ℓ is not parallel to an y of the H i and if ℓ in tersects the H i in distinct p oin ts. The follo wing prop osition sho wing the existenc e of lines in general p osition w.r.t. a p olytope illustrates a v ery useful tec hnique, the “p erturbation metho d”. The “ tr ick” b ehind this particular p erturbation metho d is that p olynomials (in one v ariable) ha v e a ﬁnite n um b er of zeros. Prop osition 7.3 L et P b e any p olytop e in E d with non empty interior. F or any two p oin ts, x and y in E d , with x outside of P ; y in the interior of P ; and x i n gener al p osition w.r.t. P , for λ ∈ R smal l en o ugh, the line, ℓ λ , thr ough x a nd y λ with y λ = y + ( λ, λ 2 , . . . , λ d ) , interse c ts P in its interior an d is in gener al p os i tion w.r.t. P . Pr o of . Assume that P is deﬁned b y t irredundan t h yp erplanes , H i , where H i is g iv en by the equation a i · z = 1 and write Λ = ( λ, λ 2 , . . . , λ d ) and u = y − x . Then the line ℓ λ is g iv en b y ℓ λ = { x + s ( y λ − x ) | s ∈ R } = { x + s ( u + Λ) | s ∈ R } . The line, ℓ λ , is not parallel to the h yp erplane H i iﬀ a i · ( u + Λ) 6 = 0 , i = 1 , . . . , t and it in tersects the H i in distinc t p oin ts iﬀ there is no s ∈ R suc h that a i · ( x + s ( u + Λ )) = 1 and a j · ( x + s ( u + Λ)) = 1 fo r some i 6 = j . Observ e that a i · ( u + Λ) = p i ( λ ) is a nonzero po lynomial in λ of degree at most d . Since a p olynomial of degree d has at most d zeros, if we let Z ( p i ) b e the (ﬁnite) set of zeros of 7.1. SHELLINGS 117 p i w e can ensure that ℓ λ is not parallel to any of the H i b y pic king λ / ∈ S t i =1 Z ( p i ) (where S t i =1 Z ( p i ) is a ﬁnite set). No w, as x is in general po sition w.r.t. P , w e ha v e a i · x 6 = 1, for i = 1 . . . , t . The condition stating that ℓ λ in tersects the H i in distinct p oin ts can b e written a i · x + sa i · ( u + Λ) = 1 and a j · x + sa j · ( u + Λ) = 1 for some i 6 = j , or sp i ( λ ) = α i and sp j ( λ ) = α j for some i 6 = j , where α i = 1 − a i · x and α j = 1 − a j · x . As x is in general p osition w.r.t. P , we ha v e α i , α j 6 = 0 and as the H i are irredundan t, the p olynomials p i ( λ ) = a i · ( u + Λ) and p j ( λ ) = a j · ( u + Λ) are not prop ortional. No w, if λ / ∈ Z ( p i ) ∪ Z ( p j ), in order for the system sp i ( λ ) = α i sp j ( λ ) = α j to ha v e a solution in s we mu st ha v e q ij ( λ ) = α i p j ( λ ) − α j p i ( λ ) = 0 , where q ij ( λ ) is not the zero p olynomial since p i ( λ ) and p j ( λ ) are not prop ortional and α i , α j 6 = 0. If w e pic k λ / ∈ Z ( q ij ), then q ij ( λ ) 6 = 0. Therefore, if w e pic k λ / ∈ t [ i =1 Z ( p i ) ∪ t [ i 6 = j Z ( q ij ) , the line ℓ λ is in g ene ral p osition w.r.t. P . Finally , w e can pic k λ small enough so that y λ = y + Λ is close enough to y so that it is in the in terior o f P . It should b e noted that the p erturbation metho d inv olving Λ = ( λ, λ 2 , . . . , λ d ) is quite ﬂexible. F or example, b y adapting the pro of of Propo sition 7.3 w e can pro v e that for an y t w o distin ct facets, F i and F j of P , there is a line in general p osition w.r.t. P inte rsecting F i and F j . Start with x outside P and v ery close to F i and y in the inte rior of P and v ery close to F j . Finally , b efore pro ving the existe nce of line shellin gs for p olytop es, w e need more t ermi- nology . Giv en an y p oin t, x , strictly o utside a p olytope, P , w e say that a facet, F , of P is visible fr om x iﬀ for ev ery y ∈ F the line through x and y interse cts F only in y (equiv alen tly , x and the in terior of P are strictly separared by the supp orting hy p erplane of F ). W e now pro v e the f o llo wing fundamen ta l theorem due to Bruggesse r and Mani [12] (1970): Theorem 7.4 (Existenc e of Line Shel lings f or Polytop e s ) L et P b e any p olytop e in E d of dimension d . F o r every p oint, x , outside P and in gener al p osition w.r.t. P , ther e is a shel l i n g of P in which the fac ets of P that ar e visible fr om x c ome ﬁrst. 118 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA ℓ z 1 z 2 z 3 F 1 F 2 F 3 F 4 Figure 7.3: Shelling a p olytop e by tra v elling along a line, ℓ Pr o of . By Prop osition 7.3, w e can ﬁnd a line, ℓ , through x suc h that ℓ is in general p osition w.r.t. P and ℓ in tersects the in terior of P . Pic k one of the t w o faces in whic h ℓ in tersects P , say F 1 , let z 1 = ℓ ∩ F 1 , and orient ℓ f rom the inside of P to z 1 . As ℓ inte rsects the suppo r t ing hy p erplanes of the facets of P in distinct p oints , w e get a linearly ordered list of these in tersection p oin ts along ℓ , z 1 , z 2 , · · · , z m , z m +1 , · · · , z s , where z m +1 is the smallest elemen t, z m is the largest eleme nt and where z 1 and z s b elong to the faces of P where ℓ in tersects P . Then, as in the example illus trated b y Figure 7.2, b y tra v elling “up ward” a lo ng the line ℓ starting fro m z 1 w e get a total ordering of the facets of P , F 1 , F 2 , . . . , F m , F m +1 , . . . , F s where F i is the facet whose supporting hyperplane cuts ℓ in z i . W e claim that the ab o v e sequence is a shellin g of P . This is pro v ed b y induction on d . F o r d = 1, P consists a line segmen t and the theorem clearly holds. Consider the in tersection ∂ F j ∩ ( F 1 ∪ · · · ∪ F j − 1 ). W e need to sho w that this is an initial segmen t of a shelling of ∂ F j . If j ≤ m , i.e., if F j b ecome visible b efore w e reac h ∞ , then the ab o ve in tersection is exactly the set of facets of F j that are visib le from z j = ℓ ∩ aﬀ ( F j ). 7.1. SHELLINGS 119 Therefore, by induction on the dimens ion, these fa cets are shellable and they form an initial segmen t of a shelling of the whole b oundary ∂ F j . If j ≥ m + 1, that is, after “passing through ∞ ” and reen tering from −∞ , the interse ction ∂ F j ∩ ( F 1 ∪ · · · ∪ F j − 1 ) is the set of non-visible facets. By rev ersing the orien tation of the line, ℓ , w e see that the facets of this in tersection are shel lable and w e get the rev ersed ordering of the facets. Finally , when w e reac h the p oin t x starting from z 1 , the facets visible from x form an initial segmen t of the shelling, as claimed. Remark: The trip along the line ℓ is often described as a r o cket ﬂight starting from the surface of P view ed as a little planet (for instance, this is the description given b y Ziegler [43] (Chapter 8)). Observ e that if w e rev erse the direction of ℓ , we obtain the r eve rsal of the original line shelling. Th us, the rev ersal of a line shelling is not only a shelling but a line shelling as w ell. W e can easily prov e the follo wing corollary: Corollary 7.5 Given any p olytop e, P , the fol lowing facts hold : (1) F or any two fac ets F an d F ′ , ther e is a shel ling of P in which F c om es ﬁrst and F ′ c om es last. (2) F or any vertex, v , o f P , ther e is a shel ling of P in which the fac ets c ontaining v f o rm an in i tial se gment of the shel lin g . Pr o of . F or (1), w e use a line in general p osition and in tersectin g F and F ′ in their in terior. F o r (2), w e pic k a po int, x , b ey ond v and pick a line in general p osition through x in tersecting the interior of P . Pic k the origin, O , in the in terior of P . A po in t , x , is b eyond v iﬀ x and O lies on diﬀeren t side s of ev ery h yperplane, H i , supp orting a facet of P con taining x but on the same side of H i for eve ry h yp erplane, H i , supp orting a facet of P not con taining x . Suc h a p oin t can b e found on a line through O and v , as the reader should c hec k. Remark: A plane triangulation , K , is a pure t w o-dimensional complex in the plane suc h that | K | is homeomorphic to a closed disk. Edelsbrunner pro v es that ev ery plane trian- gulation has a shelling and from this, that χ ( K ) = 1, where χ ( K ) = f 0 − f 1 + f 2 is the Euler-P oincar ´ e c haracteristic of K , where f 0 is the n um b er of v ertices, f 1 is the n um b er of edges and f 2 is the n umber of triangles in K (see Edels brunner [17], Chapter 3). This result is an immediate conseq uence of Corollary 7.5 if one knows ab out the stereographic pro jection map, whic h will b e discuss ed in the next Chapter. W e now ha v e all the to ols needed to prov e the famous Euler-P oincar ´ e F orm ula for Poly - top es. 120 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA 7.2 The Euler - P oin car ´ e F orm ula for P olytop es W e b egin b y deﬁning a v ery imp ortant top ological concept, the Euler-P oincar ´ e ch aracteristic of a complex. Deﬁnition 7.3 Let K b e a d -dimensi onal complex. F or ev ery i , with 0 ≤ i ≤ d , we let f i denote the n um b er of i -faces of K and w e let f ( K ) = ( f 0 , · · · , f d ) ∈ N d +1 b e the f -ve ctor asso ciated with K (if neces sary w e write f i ( K ) instead of f i ). The Euler- Poinc ar ´ e char acteristic , χ ( K ), of K is deﬁned b y χ ( K ) = f 0 − f 1 + f 2 + · · · + ( − 1) d f d = d X i =0 ( − 1) i f i . Giv en an y d -dimensional p olytope, P , the f -ve ctor asso ciated with P is the f -v ector asso- ciated with K ( P ), that is, f ( P ) = ( f 0 , · · · , f d ) ∈ N d +1 , where f i , is the nu mber of i -faces of P (= the n um b er of i -faces of K ( P ) and th us, f d = 1), and the Euler-Poinc ar ´ e char acteristic , χ ( P ), of P is deﬁned b y χ ( P ) = f 0 − f 1 + f 2 + · · · + ( − 1) d f d = d X i =0 ( − 1) i f i . Moreo v er, the f -ve c tor asso ciated with the b oundary , ∂ P , o f P is the f -v ector asso ciated with K ( ∂ P ), that is, f ( ∂ P ) = ( f 0 , · · · , f d − 1 ) ∈ N d where f i , is the n um b er of i -f a ces of ∂ P (with 0 ≤ i ≤ d − 1), and the Euler-Poinc ar´ e char a cteristic , χ ( ∂ P ), of ∂ P is deﬁned by χ ( ∂ P ) = f 0 − f 1 + f 2 + · · · + ( − 1) d − 1 f d − 1 = d − 1 X i =0 ( − 1) i f i . Observ e that χ ( P ) = χ ( ∂ P ) + ( − 1) d , since f d = 1. Remark: It is con ve nien t to set f − 1 = 1. Then, some authors, including Ziegler [43] (Chap- ter 8), deﬁne the r e duc e d Euler-Poinc ar´ e char acteristic , χ ′ ( K ), of a complex (or a p olytope), K , as χ ′ ( K ) = − f − 1 + f 0 − f 1 + f 2 + · · · + ( − 1) d f d = d X i = − 1 ( − 1) i f i = − 1 + χ ( K ) , 7.2. THE EULER-POINCAR ´ E FORMULA FOR POL YTOPES 121 i.e. , they incorp orate f − 1 = 1 into the fo rm ula. A crucial observ ation for pro ving the Euler-P oincar ´ e form ula is that the Euler-P oincar´ e c haracteristic is additiv e, whic h means that if K 1 and K 2 are an y tw o complexes suc h that K 1 ∪ K 2 is also a complex, whic h implies that K 1 ∩ K 2 is also a complex (b ecause we m ust ha v e F 1 ∩ F 2 ∈ K 1 ∩ K 2 for ev ery face F 1 of K 1 and ev ery face F 2 of K 2 ), then χ ( K 1 ∪ K 2 ) = χ ( K 1 ) + χ ( K 2 ) − χ ( K 1 ∩ K 2 ) . This follo ws immediately b ecause for an y tw o sets A and B | A ∪ B | = | A | + | B | − | A ∩ B | . T o pro v e our next theorem w e will use complete induction on N × N ordered b y the lexicographic ordering. Recall that the lexicographic ordering on N × N is deﬁned as follows : ( m, n ) < ( m ′ , n ′ ) iﬀ    m = m ′ and n < n ′ or m < m ′ . Theorem 7.6 (Euler-Poinc ar ´ e F ormula) F or every p olytop e, P , we h ave χ ( P ) = d X i =0 ( − 1) i f i = 1 ( d ≥ 0 ) , and s o , χ ( ∂ P ) = d − 1 X i =0 ( − 1) i f i = 1 − ( − 1) d ( d ≥ 1) . Pr o of . W e pro v e the following statemen t: F or ev ery d -dimensional p olytop e, P , if d = 0 then χ ( P ) = 1 , else if d ≥ 1 then for ev ery shelling F 1 , . . . , F f d − 1 , of P , for ev ery j , with 1 ≤ j ≤ f d − 1 , w e ha v e χ ( F 1 ∪ · · · ∪ F j ) =  1 if 1 ≤ j < f d − 1 1 − ( − 1) d if j = f d − 1 . W e pro ceed by complete induction on ( d, j ) ≥ (0 , 1). F or d = 0 and j = 1, the p olytope P consists of a single p oin t and so, χ ( P ) = f 0 = 1, as claimed. F o r the induction step, assu me that d ≥ 1. F o r 1 = j < f d − 1 , since F 1 is a polytop e of dimension d − 1, b y the induction h yp othesis, χ ( F 1 ) = 1, a s desired. F o r 1 < j < f d − 1 , w e ha v e χ ( F 1 ∪ · · · F j − 1 ∪ F j ) = χ j − 1 [ i =1 F i ! + χ ( F j ) − χ j − 1 [ i =1 F i ! ∩ F j ! . 122 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA Since ( d, j − 1) < ( d, j ), b y the induction h yp othesis , χ j − 1 [ i =1 F i ! = 1 and since dim( F j ) = d − 1, a g ain by the induction hy p othesis, χ ( F j ) = 0 . No w, as F 1 , . . . , F f d − 1 is a shelling and j < f d − 1 , w e ha v e j − 1 [ i =1 F i ! ∩ F j = G 1 ∪ · · · ∪ G r , for some shelling G 1 , . . . , G r , . . . , G t of K ( ∂ F j ), with r < t = f d − 2 ( ∂ F j ). The fa ct that r < f d − 2 ( ∂ F j ), i.e. , that G 1 ∪ · · · ∪ G r is not the whole b oundary of F j is a prop ert y of line shellings and also follow s from Prop osition 7.2. As dim( ∂ F j ) = d − 2, and r < f d − 2 ( ∂ F j ), b y the induction h yp othesis, w e ha ve χ j − 1 [ i =1 F i ! ∩ F j ! = χ ( G 1 ∪ · · · ∪ G r ) = 1 . Conseque ntly , χ ( F 1 ∪ · · · F j − 1 ∪ F j ) = 1 + 1 − 1 = 1 , as claimed (when j < f d − 1 ). If j = f d − 1 , then w e ha v e a complete shelling o f ∂ F f d − 1 , that is, f d − 1 − 1 [ i =1 F i ! ∩ F f d − 1 = G 1 ∪ · · · ∪ G f d − 2 ( F f d − 1 ) = ∂ F f d − 1 . As dim( ∂ F j ) = d − 2, b y the induction h yp othesis, χ ( ∂ F f d − 1 ) = χ ( G 1 ∪ · · · ∪ G f d − 2 ( F f d − 1 ) ) = 1 − ( − 1) d − 1 and it follo ws that χ ( F 1 ∪ · · · ∪ F f d − 1 ) = 1 + 1 − (1 − ( − 1) d − 1 ) = 1 + ( − 1) d − 1 = 1 − ( − 1) d , establishing the induction h yp othesis in this last case. But then, χ ( ∂ P ) = χ ( F 1 ∪ · · · ∪ F f d − 1 ) = 1 − ( − 1) d and χ ( P ) = χ ( ∂ P ) + ( − 1) d = 1 , 7.3. DEHN-SOMMER VILLE EQUA TIONS F OR SIMPLICIAL POL YTOPES 123 pro ving our theorem. Remark: Other com binatorial pro ofs of t he Euler-P oincar ´ e form ula a re giv en in Gr ¨ unbaum [24] (Chapter 8), Bo issonnat and Yvinec [8] (Chapter 7) and Ew ald [18] ( Chapter 3). Co xeter giv es a pro of very close to P oincar ´ e’s ow n pro of using notions of homolog y theory [13] (Chapter IX). W e feel that the pro of based on shellings is the most direct and one of the most elegant. Inciden tly , the ab o ve pro of of the Euler-P oincar´ e form ula is v ery close to Sc hl¨ aﬂi pro of from 1852 but Sc hl¨ aﬂi did not ha v e shellings at his disp osal so his “pro of ” had a gap. The Bruggesser-Mani pro of that p olytopes are shellable ﬁlls this gap! 7.3 Dehn-So mmerville Equations for Simplicial P olytop es and h - V ectors If a d -p olytop e, P , has the prop ert y that its f a ces are all simplices, then it is called a simplicial p ol ytop e . It is easily show n that a p olytop e is simplicial iﬀ its facets ar e simplices, in whic h case, ev ery facet has d v ertices. The p olar dual of a simplicial p olytop e is called a simple p ol ytop e . W e see immediately that ev ery v ertex of a simpl e p o lytope b elongs to d facets. F o r simplicial (and simple) p olytop es it turns out that o t her remark able equations b e- sides the Eule r-Poincar ´ e form ula hold among the num b er of i -faces. Thes e equations we re disco v ered by Dehn for d = 4 , 5 (1905) and by Sommerville in the general case (1927). Al- though it is p ossible (and not diﬃcult) t o pro ve the Dehn-Sommerv ille equations b y “double coun ting”, as in Gr ¨ un baum [24] (Chapter 9) or Boissonnat and Yvinec (Chapter 7, but b e- w are, these are the dual form ulae f o r simple p olytop es), it turns out that inste ad of using the f -v ector asso ciated with a p olytope it is preferable to use what’s kno wn as the h -ve ctor b ecause for simplicial p olytop es the h -num b ers ha v e a natural in terpretation in terms of shellings. F urthermore, the statemen t of the Dehn-Sommerville equations in terms of h - v ectors is transparen t: h i = h d − i , and the pro of is ve ry simple in terms of shellings . In t he rest of this section, w e restrict our atten tion to simplic ial complexes. In order to motiv ate h -ve ctors, w e b egin b y examining more closely the structure of the new faces that are created during a shelling when the cell F j is added to the partial shelling F 1 , . . . , F j − 1 . If K is a simplicial p olytop e and V is the set of v ertices of K , then ev ery i -face of K can b e iden tiﬁed with an ( i + 1)-subset of V (t hat is, a subset of V of cardinalit y i + 1). Deﬁnition 7.4 F or an y shelling, F 1 , . . . , F s , of a simplicial complex, K , of dimension d , for ev ery j , with 1 ≤ j ≤ s , the r estriction , R j , of the fa cet, F j , is the set of “obligatory” v ertices R j = { v ∈ F j | F j − { v } ⊆ F i , f or some i with 1 ≤ i < j } . 124 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA 1 2 3 4 5 6 Figure 7.4: A connected 1-dimensional complex, G The crucial prop ert y of the R j is that the new faces, G , added a t step j (when F j is added to the shellin g) are precisely the faces in the set I j = { G ⊆ V | R j ⊆ G ⊆ F j } . The pro of o f the ab ov e fact is left as a n exercise to the reader. But then, w e obtain a partition, { I 1 , . . . , I s } , of the set of faces of the simplicial complex (other that K itself ). Note that the empt y face is allo w ed. Now, if we deﬁne h i = |{ j | | R j | = i, 1 ≤ j ≤ s }| , for i = 0 , . . . , d , then it turns out that w e can reco ve r the f k in terms of the h i as follo ws: f k − 1 = s X j =1  d − | R j | k − | R j |  = k X i =0 h i  d − i k − i  , with 1 ≤ k ≤ d . But more is true: The ab o v e equations are inv ertible a nd the h k can be expressed in terms of the f i as follo ws: h k = k X i =0 ( − 1) k − i  d − i d − k  f i − 1 , with 0 ≤ k ≤ d (remem b er, f − 1 = 1). Let us explain all this in more detail. Consider the example of a connected graph (a simplicial 1-dimensional complex ) from Ziegler [43] (Section 8.3) sho wn in Figure 7.4: A shelling order of its 7 edges is giv en b y the sequence 12 , 13 , 34 , 35 , 45 , 36 , 56 . The partial order of the faces of G to g ethe r with the blo c ks of the partition { I 1 , . . . , I 7 } asso ciated with the sev en edges of G are sho wn in Figure 7.5, with the blo c ks I j sho wn in b oldface: 7.3. DEHN-SOMMER VILLE EQUA TIONS F OR SIMPLICIAL POL YTOPES 125 ∅ 1 2 3 4 5 6 12 13 34 35 45 3 6 56 Figure 7.5: the partition asso ciated with a shelling of G The “minimal” new faces (corresp onding to the R j ’s) added a t ev ery stage of the shelling are ∅ , 3 , 4 , 5 , 45 , 6 , 56 . Again, if h i is the n um b er of blo c ks, I j , suc h that the corresp onding restriction set, R j , has size i , that is, h i = |{ j | | R j | = i, 1 ≤ j ≤ s }| , for i = 0 , . . . , d , where the simplicial p olytop e, K , has dimension d − 1, we deﬁne the h -ve c tor asso ciated with K as h ( K ) = ( h 0 , . . . , h d ) . Then, in the ab o v e example, as R 1 = {∅} , R 2 = { 3 } , R 3 = { 4 } , R 4 = { 5 } , R 5 = { 4 , 5 } , R 6 = { 6 } and R 7 = { 5 , 6 } , w e get h 0 = 1, h 1 = 4 and h 2 = 2, that is, h ( G ) = (1 , 4 , 2) . No w, let us sho w that if K is a shellable sim plicial complex, then the f -v ector can b e reco v ered f r o m the h -v ector. Indeed, if | R j | = i , then each ( k − 1)-face in the blo c k I j m ust use all i no des in R j , so that there are only d − i no des a v ailable and, among those, k − i m ust b e c hosen. Therefore, f k − 1 = s X j =1  d − | R j | k − | R j |  and, b y deﬁnition of h i , w e get f k − 1 = k X i =0 h i  d − i k − i  = h k +  d − k + 1 1  h k − 1 + · · · +  d − 1 k − 1  h 1 +  d k  h 0 , ( ∗ ) where 1 ≤ k ≤ d . Moreo v er, the formulae are in v ertible, that is, the h i can b e expressed in terms of the f k . F or this, form the t w o p olynomials f ( x ) = d X i =0 f i − 1 x d − i = f d − 1 + f d − 2 x + · · · + f 0 x d − 1 + f − 1 x d 126 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA with f − 1 = 1 and h ( x ) = d X i =0 h i x d − i = h d + h d − 1 x + · · · + h 1 x d − 1 + h 0 x d . Then, it is easy to see that f ( x ) = d X i =0 h i ( x + 1 ) d − i = h ( x + 1) . Conseque ntly , h ( x ) = f ( x − 1) and b y comparing the co eﬃc ien ts of x d − k on b oth sides of the ab o v e equation, w e get h k = k X i =0 ( − 1) k − i  d − i d − k  f i − 1 . In particular, h 0 = 1, h 1 = f 0 − d , and h d = f d − 1 − f d − 2 + f d − 3 + · · · + ( − 1) d − 1 f 0 + ( − 1) d . It is also easy to c hec k that h 0 + h 1 + · · · + h d = f d − 1 . No w, w e just sho w ed that if K is she llable, then its f -v ector a nd its h -v ector a re related as abov e. But ev en if K is not shellable, the ab o ve suggests deﬁning the h - v ector from the f -ve ctor as a bov e. Th us, w e mak e the deﬁnition: Deﬁnition 7.5 F or an y ( d − 1)-dimensional simplicial complex, K , the h -ve ctor associated with K is the v ector h ( K ) = ( h 0 , . . . , h d ) ∈ Z d +1 , giv en by h k = k X i =0 ( − 1) k − i  d − i d − k  f i − 1 . Note that if K is shellable, then the inte rpretation of h i as the n um b er of cells, F j , suc h that the corresponding restric tion set, R j , has size i show s that h i ≥ 0. How ev er, for an arbitrary simplicial complex, some of the h i can b e strictly negativ e. Suc h an example is giv en in Ziegler [43] (Section 8.3). W e summarize below most of what we just sho we d: 7.3. DEHN-SOMMER VILLE EQUA TIONS F OR SIMPLICIAL POL YTOPES 127 Prop osition 7.7 L et K b e a ( d − 1) -dimens i o nal pur e si m plicial c omplex. If K is shel lable, then its h -ve ctor is nonne gative and h i c ounts the numb er of c el ls in a shel ling whose r estric- tion set has size i . Mor e over, the h i do no t dep end on the p articular shel ling of K . There is a w ay of computing the h -v ector of a pure simplicial complex fro m its f -v ector reminisce nt of the P ascal triangle (except that negativ e en tries can turn up). Again, the reader is referred to Ziegler [43] (Section 8.3). W e are now ready to prov e the Dehn-Sommerville equations . F or d = 3, these are easily obtained by double countin g. Indeed, for a simplici al p olytop e, ev ery edge b elongs to t w o facets and ev ery facet has three edges. It fo llo ws tha t 2 f 1 = 3 f 2 . T ogether with Euler’s formu la f 0 − f 1 + f 2 = 2 , w e see that f 1 = 3 f 0 − 6 a nd f 2 = 2 f 0 − 4 , namely , that the n um b er of v ertices of a simplicial 3-p olytop e determines its n um b er of edges and faces, these b eing linear functions of the n um b er of v ertice s. F or a rbitra r y dimension d , w e hav e Theorem 7.8 (Dehn-Som mervil le Equations) If K is any simplicial d -p o lytop e, then the c om p onents of the h -ve c tor satisfy h k = h d − k k = 0 , 1 , . . . , d. Equivalently f k − 1 = d X i = k ( − 1) d − i  i k  f i − 1 k = 0 , . . . , d. F urthermor e, the e quation h 0 = h d is e quivalent to the Euler-Poinc ar ´ e formula. Pr o of . W e presen t a short and elegan t pro of due to McMullen . Recall from Prop osition 7.2 that the rev ersal, F s , . . . , F 1 , of a shell ing, F 1 , . . . , F s , of a p olytop e is also a shelling. F ro m this, we see that for ev ery F j , the restriction set of F j in the rev ersed shelling is equal to R j − F j , t he complem ent of the restriction set of F j in the original shelling. Therefore, if | R j | = k , then F j con tributes “1” to h k in the original shelling iﬀ it con tributes “1” to h d − k in the rev ersed shelling (where | R j − F j | = d − k ). It follo ws that the v alue of h k computed in the original shelling is the same as the v alue of h d − k computed in the rev ersed shelling. Ho w ev er, b y Prop osition 7.7 , the h -ve ctor is indep enden t of the shellin g and hence, h k = h d − k . 128 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA Deﬁne the p olynomials F ( x ) and H ( x ) b y F ( x ) = d X i =0 f i − 1 x i ; H ( x ) = (1 − x ) d F  x 1 − x  . Note that H ( x ) = P d i =0 f i − 1 x i (1 − x ) d − i and an easy computation sho ws that the co eﬃcie nt of x k is equal to k X i =0 ( − 1) k − i  d − i d − k  f i − 1 = h k . No w, the equations h k = h d − k are equiv alen t to H ( x ) = x d H ( x − 1 ) , that is, F ( x − 1) = ( − 1) d F ( − x ) . As F ( x − 1 ) = d X i =0 f i − 1 ( x − 1) i = d X i =0 f i − 1 i X j =0  i i − j  x i − j ( − 1) j , w e see that the co eﬃcien t of x k in F ( x − 1 ) (obtained when i − j = k , that is, j = i − k ) is d X i =0 ( − 1) i − k  i k  f i − 1 = d X i = k ( − 1) i − k  i k  f i − 1 . On the other hand, the coeﬃcien t of x k in ( − 1) d F ( − x ) is ( − 1) d + k f k − 1 . By equating the co eﬃcie nts of x k , w e get ( − 1) d + k f k − 1 = d X i = k ( − 1) i − k  i k  f i − 1 , whic h, b y multiply ing b oth sides by ( − 1) d + k , is equiv alen t to f k − 1 = d X i = k ( − 1) d + i  i k  f i − 1 = d X i = k ( − 1) d − i  i k  f i − 1 , as claimed. Finally , a s we already kno w that h d = f d − 1 − f d − 2 + f d − 3 + · · · + ( − 1) d − 1 f 0 + ( − 1) d and h 0 = 1, b y m ultiplying b oth sides of the equation h d = h 0 = 1 b y ( − 1) d − 1 and mov ing ( − 1) d ( − 1) d − 1 = − 1 to the right hand side, we get the Euler-P oincar´ e formula. 7.3. DEHN-SOMMER VILLE EQUA TIONS F OR SIMPLICIAL POL YTOPES 129 Clearly , the D ehn-Sommerville equations, h k = h d − k , are linearly independen t for 0 ≤ k < ⌊ d +1 2 ⌋ . F o r example, for d = 3, w e hav e the t w o indep enden t equations h 0 = h 3 , h 1 = h 2 , and for d = 4, w e also ha v e tw o independen t equations h 0 = h 4 , h 1 = h 3 , since h 2 = h 2 is trivial. When d = 3, w e know that h 1 = h 2 is equiv alen t to 2 f 1 = 3 f 2 and when d = 4, if o ne unrav els h 1 = h 3 in terms of the f i ’ one ﬁnds 2 f 2 = 4 f 3 , that is f 2 = 2 f 3 . More generally , it is easy to c hec k that 2 f d − 2 = d f d − 1 for all d . F or d = 5, we ﬁnd three indep enden t equations h 0 = h 5 , h 1 = h 4 , h 2 = h 3 , and so on. It can b e sho wn that for g ene ral d - polytop es, the Euler-P oincar´ e formu la is the only equation satisﬁed by all h - v ectors a nd for simplicial d -p olytop es, the ⌊ d +1 2 ⌋ D ehn-Somme rville equations, h k = h d − k , are the only equations satisﬁed b y all h -v ectors (see Gr ¨ un baum [24], Chapter 9). Remark: Readers familiar with homolo g y and cohomology may susp ect that the Dehn- Sommerville equations are a consequence of a ty p e of Poinc ar´ e dualit y . Stanley pro v ed that this is indeed the case. It turns out that the h i are the dimensions of cohomology groups o f a certain toric variety a sso ciated with the p olytop e. F o r more on this topic, see Stanley [35] (Chapters I I and II I) and F ulton [19 ] (Section 5 .6 ) . As w e sa w f or 3-dimensional simplicial p olytop es, the n um b er of v ertices, n = f 0 , de- termines the nu mber of edges and the n um b er of faces, and these are linear in f 0 . F or d ≥ 4, this is no longer true and the num b er of facets is no longer linear in n but in fact quadratic. It is then natural to ask whic h d -p olytop es with a prescribed num b er o f v ertices ha v e the maxim um num b er of k -faces. This question whic h remained an op en problem for some t w en t y y ears was ev en tually settled b y McMulle n in 1970 [27]. W e will presen t this result (without pro of ) in the next sec tion. 130 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA 7.4 The Upp er Bound The o rem and Cyclic P olytop es Giv en a d -p olytop e with n vertic es, what is an upp er b ound on the n um b er of its i -faces? This question is not only import a n t from a theoretical p oin t of vie w but also from a computational p oin t of view b ecause of its implications for algorithms in com binatorial optimization and in computational geometry . The answ er to the ab o v e problem is that there is a class of p olytop es called cyclic p olytop es suc h that the cyclic d -p olytop e, C d ( n ), has the maxim um num b er of i -faces a mong all d - p olytopes with n vertic es. This result stated by Motzkin in 1957 b ecame kno wn as the upp er b ound c onje ctur e un til it w as pro v ed by McMullen in 1970, using shellings [27] (just after Bruggesser and Mani’s pro of that p olytopes are shellable). It is no w kno wn as the upp er b ound the or em . Another pro o f of the upp er b ound theorem w as g iv en later b y Alon and Kalai [2] (1985). A v ersion of this pro of can also b e found in Ew ald [18] (Chapter 3). McMullen’ s pro of is not really ve ry diﬃc ult but it is still quite in v olv ed so w e will only state some prop ositions needed for its pro o f. W e urge the reader to read Ziegler’s accoun t of this b eautiful pro of [43] (Chapter 8). W e b egin with cyclic p olytopes. First, consider the cases d = 2 a nd d = 3. When d = 2, our p olytop e is a p olygon in whic h case n = f 0 = f 1 . Th us, this case is trivial. F o r d = 3 , w e claim that 2 f 1 ≥ 3 f 2 . Indeed, ev ery edge b elongs to exactly t w o faces so if w e add up the n um b er of sides for all faces, w e get 2 f 1 . Since ev ery face has at least three sides, w e get 2 f 1 ≥ 3 f 2 . Then, using Euler’s relation, it is easy to sho w that f 1 ≤ 6 n − 3 f 2 ≤ 2 n − 4 and w e kno w that equalit y is ac hiev ed for simplicial p olytop es. Let us now consider the general case. The ra tional curve , c : R → R d , giv en parametrically b y c ( t ) = ( t, t 2 , . . . , t d ) is at the heart of the story . This curv e if often called the moment curve or r ational normal curve of degree d . F or d = 3, it is kno wn as the twiste d cubic . Here is the deﬁnition of t he cyclic p olytope, C d ( n ). Deﬁnition 7.6 F or an y seque nce, t 1 < . . . < t n , of distinct real nu mber, t i ∈ R , with n > d , the con v ex hull, C d ( n ) = con v( c ( t 1 ) , . . . , c ( t n )) of the n p oin ts, c ( t 1 ) , . . . , c ( t n ), on the momen t curv e of degree d is called a cyclic p o lytop e . The ﬁrst in teresting fact ab out the cyclic p olytop e is that it is simplicial. Prop osition 7.9 Every d + 1 o f the p oints c ( t 1 ) , . . . , c ( t n ) ar e aﬃnely indep endent. Conse- quently, C d ( n ) is a simplic ial p olytop e and the c ( t i ) ar e vertic e s . 7.4. THE UPPER BO UND THEOREM 131 Pr o of . W e ma y a ss ume that n = d + 1. Sa y c ( t 1 ) , . . . , c ( t n ) b elong to a hyperplane, H , giv en b y α 1 x 1 + · · · + α d x d = β . (Of course, not all the α i are zero.) Then, we hav e the po lyn omial, H ( t ), giv en b y H ( t ) = − β + α 1 t + α 2 t 2 + · · · + α d t d , of degree a t most d and as eac h c ( t i ) belong to H , w e se e that eac h c ( t i ) is a zero of H ( t ). Ho w ev er, there are d + 1 distinct c ( t i ), so H ( t ) would ha v e d + 1 distinc t ro ots. As H ( t ) has degree at most d , it m ust b e the zero p olynomial, a con tradiction. Returing to the original n > d + 1, w e just pro v ed ev ery d + 1 of t he p oints c ( t 1 ) , . . . , c ( t n ) are aﬃnely indep enden t. Then, ev ery prop er face of C d ( n ) has at most d indep enden t v ertices, whic h means that it is a simplex . The follow ing prop osition already sho ws that the cyclic p olytope, C d ( n ), has  n k  ( k − 1)- faces if 1 ≤ k ≤ ⌊ d 2 ⌋ . Prop osition 7.10 F or any k w i th 2 ≤ 2 k ≤ d , eve ry subset o f k vertic es of C d ( n ) is a ( k − 1) -fac e of C d ( n ) . Henc e f k ( C d ( n )) =  n k + 1  if 0 ≤ k <  d 2  . Pr o of . Consider any sequence t i 1 < t i 2 < · · · < t i k . W e will pro v e that there is a h yp erplane separating F = con v( { c ( t i 1 ) , . . . , c ( t i k ) } ) and C d ( n ). Consider the p olynomial p ( t ) = k Y j =1 ( t − t i j ) 2 and write p ( t ) = a 0 + a 1 t + · · · + a 2 k t 2 k . Consider the v ector a = ( a 1 , a 2 , . . . , a 2 k , 0 , . . . , 0) ∈ R d and the h yp erplane, H , giv en by H = { x ∈ R d | x · a = − a 0 } . Then, for eac h j with 1 ≤ j ≤ k , w e hav e c ( t i j ) · a = a 1 t i j + · · · + a 2 k t 2 k i j = p ( t i j ) − a 0 = − a 0 , and so, c ( t i j ) ∈ H . On the other hand, for an y other p oin t, c ( t i ), distinct f rom an y of the c ( t i j ), w e ha v e c ( t i ) · a = − a 0 + p ( t i ) = − a 0 + k Y j =1 ( t i − t i j ) 2 > − a 0 , 132 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA pro ving that c ( t i ) ∈ H + . But then, H is a supporting h yperplane of F for C d ( n ) and F is a ( k − 1)-face. Observ e that Prop osition 7.10 sho ws that an y subset of ⌊ d 2 ⌋ v ertices of C d ( n ) forms a face of C d ( n ). When a d -p olytop e has this prop ert y it is called a neigh b orly p olytop e . Therefore, cyclic p olytop es are neigh b orly . Prop osition 7.10 also sho ws a phenomenon that only manifests itself in dimension at least 4: F or d ≥ 4, the p o lytope C d ( n ) has n pairwise adjacen t v ertices. F or n >> d , this is coun ter-intuitiv e. Finally , the com binatorial structure of cyclic p olytopes is complete ly determined as fol- lo ws: Prop osition 7.11 (Gale evenness c ondition , Gale (1963)). L et n and d b e inte gers with 2 ≤ d < n . F or any se quenc e t 1 < t 2 < · · · < t n , c onsider the cyclic p olytop e C d ( n ) = con v( c ( t 1 ) , . . . , c ( t n )) . A subset, S ⊆ { t 1 , . . . , t n } with | S | = d determin es a fac et of C d ( n ) iﬀ for al l i < j not in S , then the numb er of k ∈ S b etwe en i and j is ev e n: |{ k ∈ S | i < k < j }| ≡ 0 (mo d 2) for i, j / ∈ S Pr o of . W rite S = { s 1 , . . . , s d } ⊆ { t 1 , . . . , t n } . Consider the p oly omial q ( t ) = d Y i =1 ( t − s i ) = d X j =0 b j t j , let b = ( b 1 , . . . , b d ), and let H b e the h yp erplane giv en b y H = { x ∈ R d | x · b = − b 0 } . Then, for eac h i , with 1 ≤ i ≤ d , w e ha v e c ( s i ) · b = d X j =1 b j s j i = q ( s i ) − b 0 = − b 0 , so that c ( s i ) ∈ H . F or all other t 6 = s i , q ( t ) = c ( t ) · b + b 0 6 = 0 , that is, c ( t ) / ∈ H . There fore, F = { c ( s 1 ) , . . . , c ( s d ) } is a fa cet of C d ( n ) iﬀ { c ( t 1 ) , . . . , c ( t n ) } − F lies in one of the tw o op en half-spaces determined b y H . This is equiv alen t to q ( t ) c hanging its sign an ev en n um b er of times while, increasing t , w e pass through the ve rtices in F . Therefore, the prop osition is pro v ed. In particular, Prop osition 7.11 sho ws that the com binatorial structure of C d ( n ) do es not dep end on the speciﬁc choice of the sequ ence t 1 < · · · < t n . This j ustiﬁes o ur notation C d ( n ). Here is the celebrated upp er b ound theorem ﬁrst prov ed b y McMullen [27]. 7.4. THE UPPER BO UND THEOREM 133 Theorem 7.12 (Upp e r Bound The or em, McMul len (1970)) L e t P b e any d -p ol ytop e with n vertic es. The n, for every k , with 1 ≤ k ≤ d , the p olytop e P has a t most as many ( k − 1) -fac es as the cyclic p olytop e, C d ( n ) , that is f k − 1 ( P ) ≤ f k − 1 ( C d ( n )) . Mor e over, e quality for s o me k with ⌊ d 2 ⌋ ≤ k ≤ d implies that P is neighb orly. The ﬁrst step in the pro of of Theorem 7.12 is to prov e that among all d - polytop es with a giv en num b er, n , of v ertices, the maxim um num b er of i -faces is ac hiev ed by simplicial d -p olytop es. Prop osition 7.13 Given any d -p olytop e, P , with n -vertic es, it is p ossible to form a simpli- cial p olytop e, P ′ , by p erturbing the vertic es of P such that P ′ also h as n vertic es and f k − 1 ( P ) ≤ f k − 1 ( P ′ ) for 1 ≤ k ≤ d. F urthermor e, e quality for k > ⌊ d 2 ⌋ c an o c cur only if P is simplicial. Sketch of p r o of . First, w e apply Prop osition 6 .8 to tria ng ulate the facets of P without adding an y ve rtices. Then, w e can p erturb the v ertices to obtain a simplicial p olytop e, P ′ , with at least as man y facets (and th us, faces) as P . Prop osition 7.13 allo ws us to restict our atten tion to simpl icial p olytop es. No w, it is ob vious that f k − 1 ≤  n k  for an y p olytope P (simplicial or not) and w e also kno w that equalit y holds if k ≤ ⌊ d 2 ⌋ for neigh b orly p olytop es suc h as the cyclic polytop es. F or k > ⌊ d 2 ⌋ , it turns out that equalit y can only b e ac hiev ed for simplices. Ho w ev er, for a simplicial p olytop e, the Dehn-Sommervill e equations h k = h d − k together with the equations ( ∗ ) g iving f k in terms of the h i ’s sho w that f 0 , f 1 , . . . , f ⌊ d 2 ⌋ already deter- mine the whole f -v ector. Th us, it is p ossible to expres s t he f k − 1 in terms of h 0 , h 1 , . . . , h ⌊ d 2 ⌋ for k ≥ ⌊ d 2 ⌋ . It turns out that w e get f k − 1 = ⌊ d 2 ⌋ X ∗ i =0  d − i k − i  +  i k − d + i  h i , where the meaning of the superscript ∗ is that when d is eve n w e only tak e half of the last term for i = d 2 and when d is o dd w e take the whole last term for i = d − 1 2 (for details, see Ziegler [43], Chapter 8). As a consequence if we can sho w that the neigh b orly p olytop es maximize not only f k − 1 but also h k − 1 when k ≤ ⌊ d 2 ⌋ , the upp er b ound theorem will b e pro v ed. Indeed , McMullen prov ed the f ollo wing theorem whic h is “more than enough” to yield the desired result ([27]): 134 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA Theorem 7.14 (McMul len (1970) ) F or every simplicial d - p olytop e with f 0 = n vertic es, we have h k ( P ) ≤  n − d − 1 + k k  for 0 ≤ k ≤ d. F urthermor e, e quality holds for al l l and al l k wi th 0 ≤ k ≤ l iﬀ l ≤ ⌊ d 2 ⌋ and P is l -neighb o rly. (a p olytop e is l -neighb o rly iﬀ any subset of l or less vertic es determine a fac e of P .) The pro of of Theorem 7.14 is to o in v olve d to b e g iv en here, whic h is unfortunate, since it is really b eautiful. It mak es a clev er use of shellings and a careful analysis of the h -num b ers of links of v ertices. Aga in, the reader is referred to Ziegler [43], Chapter 8. Since cyclic d -p olytop es are neigh b orly (whic h means that they are ⌊ d 2 ⌋ -neigh b orly), The- orem 7.12 follo ws from Prop osition 7.13, and Theorem 7.14. Corollary 7.15 F or ev e ry simplicia l neighb o rly d -p o l ytop e w ith n vertic es, we have f k − 1 = ⌊ d 2 ⌋ X ∗ i =0  d − i k − i  +  i k − d + i   n − d − 1 + i i  for 1 ≤ k ≤ d. This gi v es the maxim um numb er of ( k − 1) -f a c es f o r any d -p olytop e with n -vertic es, for al l k with 1 ≤ k ≤ d . In p articular, the n umb er of fa c ets of the cyclic p olytop e, C d ( n ) , is f d − 1 = ⌊ d 2 ⌋ X ∗ i =0 2  n − d − 1 + i i  and, mor e explicitly, f d − 1 =  n − ⌊ d +1 2 ⌋ n − d  +  n − ⌊ d +2 2 ⌋ n − d  . Corollary 7.15 implies that the n umber o f facets of an y d - polytop e is O ( n ⌊ d 2 ⌋ ). An unfor- tunate consequence of this upp er b ound is that the complexit y of an y con v ex hull algorithms for n p oin ts in E d is O ( n ⌊ d 2 ⌋ ). The O ( n ⌊ d 2 ⌋ ) upp er b ound can b e obtained more directly using a prett y argumen t using shellings due to R. Seidel [34]. Consider an y shelli ng of an y simp licial d -p olytop e, P . F or ev ery facet, F j , of a shelling either the restriction set R j or its complemen t F j − R j has a t most ⌊ d 2 ⌋ eleme nts. So, either in the shelling or in the rev ersed shelling, the restriction set o f F j has at most ⌊ d 2 ⌋ elemen ts. Moreo v er, the restriction sets are all distinct, b y construction. Th us, the nu mber of facets is at most t wice the n um b er of k -faces of P with k ≤ ⌊ d 2 ⌋ . It follo ws that f d − 1 ≤ 2 ⌊ d 2 ⌋ X i =0  n i  7.4. THE UPPER BO UND THEOREM 135 and this rough estimate yields a O ( n ⌊ d 2 ⌋ ) b ound. Remark: There is also a lower b ound the or em due to Ba r nette (197 1 , 1973) whic h giv es a lo w er b ound on the f - v ectors all d -p olytop es with n v ertices. In this case, there is an analog of the cyclic p olytop es called stacke d p olytop es . These p olytop es, P d ( n ), are simplicial p olytop es obtained from a simplex by building shallow p yramids o v er the facets of the simplex. Then, it turns out that if d ≥ 2, then f k ≥   d k  n −  d +1 k +1  k if 0 ≤ k ≤ d − 2 ( d − 1 ) n − ( d + 1)( d − 2) if k = d − 1. There has b een a lot of progress on the com binatorics of f -v ectors and h -v ectors since 1971, esp ecially by R. Stanley , G . Kalai and L. Billera and K. Lee, among others. W e recommend t w o exc ellen t surv eys: 1. Bay er and Lee [4 ] summ arizes progress in this area up to 1993. 2. Billera and Bj¨ orner [7] is a more adv anced surv ey whic h repo r ts on results up to 1997. In fa ct, man y o f the c hapters in Go o dman a nd O’Rourk e [22] should b e o f in terest to the reader. Generalizations of the Upp er Bound Theorem using sophisticated tec hniques (face rings) due to Stanley can b e found in Stanley [35] (Chapters II) and connections with toric v arieties can b e found in Sta nley [35] (Chapters I I I) and F ulton [19]. 136 CHAPTER 7. SHELLINGS AND THE EULER-POINCAR ´ E FORMULA Chapter 8 Diric hlet–V oronoi Diagram s and Delauna y T riangulations 8.1 Diric hlet–V oronoi Diagrams In this c hapter w e presen t v ery brieﬂy the concepts of a V or o noi diagr a m a nd o f a Delauna y triangulation. These are imp ortant to ols in computational geometry , and Delauna y triangu- lations are imp ortan t in problems whe re it is neces sary to ﬁt 3D data using surface splines. It is usually useful to compute a go o d mesh for the pro jection of this set of data p oin ts on to the xy -plane, and a Delaunay triangulation is a g o o d candidate. Our pres en tation will b e rather sk etc hy . W e a re primarily intere sted in deﬁning these concepts and stating their most imp ortan t prop erties. F o r a comprehensi ve exp osition o f V oronoi diagrams, Delauna y trian- gulations, and more topics in computational geometry , our reade rs may consul t O’Rourke [29], Preparata a nd Shamos [30], Boissonnat and Yvinec [8], de Berg, V an Krev eld, Ov er- mars, and Sc h w arzk opf [5], or R isle r [31]. The surv ey by Graham and Y ao [23 ] con tains a v ery gen tle and lucid in tro duction to computational geometry . Some practical applications of V oronoi diagrams and D elaunay triangulatio ns are brieﬂy discusse d in Section 8.7. Let E b e a Euclidean space of ﬁnite dimension, that is, an aﬃne space E whose underlying v ector space − → E is equipped with an inner pro duct (and has ﬁnite dimension). F o r concrete- ness, one ma y safely assume that E = E m , although what follo ws applies to an y Euclidean space of ﬁnite dimension. Giv en a set P = { p 1 , . . . , p n } of n p oin ts in E , it is often useful to ﬁnd a partition of the space E into regions eac h con taining a single po int of P and having some nice prop erties . It is also often useful to ﬁnd triangulations of the con v ex h ull of P ha ving some nice prop erties. W e shall see that this can b e done and that the t w o problems are closely related. In order to solv e the ﬁrst problem, w e need to in tro duce bise ctor lines and bisector planes. F o r simplic ity , let us ﬁrst assume that E is a plane i.e., has dimension 2 . Give n an y t w o distinct p oin ts a, b ∈ E , the line orthogonal to the line segmen t ( a, b ) and passing through the midpo int of this segmen t is the lo cus o f all p oin ts ha ving equal distance to a and b . It 137 138 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS b b L a b Figure 8.1: The bisector line L of a and b is called the bise ctor line of a and b . The bise ctor line of t w o po in ts is illustrated in Figure 8.1. If h = 1 2 a + 1 2 b is the midpoint of the line segmen t ( a, b ), letting m b e an arbitrary p oint on the bisector line, the equation of this line can b e found b y writing that hm is orthogonal to ab . In any orthogonal frame, letting m = ( x, y ), a = ( a 1 , a 2 ), b = ( b 1 , b 2 ), the equation of this line is ( b 1 − a 1 )( x − ( a 1 + b 1 ) / 2) + ( b 2 − a 2 )( y − ( a 2 + b 2 ) / 2) = 0 , whic h can also b e written as ( b 1 − a 1 ) x + ( b 2 − a 2 ) y = ( b 2 1 + b 2 2 ) / 2 − ( a 2 1 + a 2 2 ) / 2 . The closed half-plane H ( a, b ) con taining a and with b oundary the bisector line is the lo cus of all p oin ts suc h that ( b 1 − a 1 ) x + ( b 2 − a 2 ) y ≤ ( b 2 1 + b 2 2 ) / 2 − ( a 2 1 + a 2 2 ) / 2 , and the closed half - plane H ( b, a ) con taining b and with b oundary the bisector line is the lo cus of a ll p oints suc h that ( b 1 − a 1 ) x + ( b 2 − a 2 ) y ≥ ( b 2 1 + b 2 2 ) / 2 − ( a 2 1 + a 2 2 ) / 2 . The closed half- plane H ( a, b ) is the set of all p oin ts whose distance to a is less that or equal to the distance to b , and vice v ersa for H ( b, a ). Th us, p oin ts in the closed half-plane H ( a, b ) are closer to a than they are to b . W e no w consider a problem called the p ost oﬃc e pr oblem by Graham and Y ao [23]. Giv en an y set P = { p 1 , . . . , p n } of n p oints in the plane (considered as p ost oﬃc es or sites ), for 8.1. DIRICHLET–V ORONOI DIA GRAMS 139 an y arbitrary p oint x , ﬁnd out whic h post oﬃce is closest to x . Since x can b e arbitrary , it seems desirable to precompute the sets V ( p i ) consisting of all p oin ts that are closer to p i than to any other p oin t p j 6 = p i . Indeed, if the sets V ( p i ) are kno wn, the answ er is an y po st oﬃce p i suc h that x ∈ V ( p i ). Thu s, it remains to compute the sets V ( p i ). F or this, if x is closer to p i than to any other p oin t p j 6 = p i , then x is on the same side as p i with resp ect to the bisector line of p i and p j for ev ery j 6 = i , and th us V ( p i ) = \ j 6 = i H ( p i , p j ) . If E has dimension 3, the lo cus of all p oin ts ha ving equal distanc e to a and b is a plane. It is called the bise ctor plane of a and b . The equation of this plane is also found b y writing that hm is orthogonal to ab . The equation of this plane is ( b 1 − a 1 )( x − ( a 1 + b 1 ) / 2) + ( b 2 − a 2 )( y − ( a 2 + b 2 ) / 2) + ( b 3 − a 3 )( z − ( a 3 + b 3 ) / 2) = 0 , whic h can also b e written as ( b 1 − a 1 ) x + ( b 2 − a 2 ) y + ( b 3 − a 3 ) z = ( b 2 1 + b 2 2 + b 2 3 ) / 2 − ( a 2 1 + a 2 2 + a 2 3 ) / 2 . The closed half-space H ( a, b ) con taining a and with b oundary the bisector plane is the lo cus of all p oin ts suc h that ( b 1 − a 1 ) x + ( b 2 − a 2 ) y + ( b 3 − a 3 ) z ≤ ( b 2 1 + b 2 2 + b 2 3 ) / 2 − ( a 2 1 + a 2 2 + a 2 3 ) / 2 , and the closed half-space H ( b, a ) con taining b and with b oundary the bisector plane is t he lo cus of a ll p oints suc h that ( b 1 − a 1 ) x + ( b 2 − a 2 ) y + ( b 3 − a 3 ) z ≥ ( b 2 1 + b 2 2 + b 2 3 ) / 2 − ( a 2 1 + a 2 2 + a 2 3 ) / 2 . The closed half- space H ( a, b ) is the set of all p oin ts whose distance to a is less t hat or equal to the distance to b , and vice v ersa for H ( b, a ). Again, p oints in the closed half-space H ( a, b ) are closer to a than they are to b . Giv en an y set P = { p 1 , . . . , p n } of n po ints in E (of dimension m = 2 , 3), it is often useful to ﬁnd fo r ev ery p oint p i the region consis ting o f all p oin ts that are closer to p i than to an y other p oin t p j 6 = p i , that is, the set V ( p i ) = { x ∈ E | d ( x, p i ) ≤ d ( x, p j ) , for all j 6 = i } , where d ( x, y ) = ( xy · xy ) 1 / 2 , the Euclidean distance a sso ciated with the inner pro duct · on E . F ro m the deﬁnition of the bisector line (or plane), it is immediate that V ( p i ) = \ j 6 = i H ( p i , p j ) . 140 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS F a milies of sets of the form V ( p i ) we re inv estigated by Diric hlet [15] (1850) and V oronoi [42] (1908). V oronoi diagrams also arise in crystallograph y (Gilb ert [21]). Other applications, including facilit y lo cation and path planning, are discu ssed in O’Rourk e [29]. F or simplicit y , w e also denote the set V ( p i ) b y V i , and w e in tro duce the following deﬁnition. Deﬁnition 8.1 Let E b e a Euclidean space of dimens ion m ≥ 1. Giv en an y set P = { p 1 , . . . , p n } of n p oin ts in E , the Dirichle t–V or onoi dia g r am V or ( P ) of P = { p 1 , . . . , p n } is the family of subsets of E consis ting of the sets V i = T j 6 = i H ( p i , p j ) and of all of their in tersections. Diric hlet–V oro noi diagrams are also called V or onoi dia gr ams , V or onoi tessel lations , or Thiessen p olygons . F o llowing common usage, w e will use the terminology V or onoi diagr a m . As in tersections of con v ex sets (closed half-planes or closed half - space s), the V or onoi r e g ions V ( p i ) are conv ex sets. In dimension tw o, the bo undaries of these regions are conv ex p olygons, and in dimens ion three, the b oundaries a r e conv ex p olyhedra. Whether a region V ( p i ) is b o unde d or not dep ends on the lo cation of p i . If p i b elongs to the bo undary of the conv ex h ull of the set P , then V ( p i ) is un b ounded, and otherwise b ounded. In dimension t w o, the con v ex h ull is a con v ex p olygon, and in dimens ion three, the con ve x h ull is a con v ex p olyhedron. As w e will see later, there is an in timate relationship b et w een conv ex hulls and V oronoi diagrams. Generally , if E is a Euclidean space of dimension m , giv en any t wo distinct p oin ts a, b ∈ E , the lo cus o f a ll p oin ts having equal distance to a and b is a hyperplane. It is called the bis e ctor hyp erpla ne of a and b . The equation of this hyperplane is still found b y writing that hm is orthogonal to ab . The equation of this h yp erplane is ( b 1 − a 1 )( x 1 − ( a 1 + b 1 ) / 2) + · · · + ( b m − a m )( x m − ( a m + b m ) / 2) = 0 , whic h can also b e written as ( b 1 − a 1 ) x 1 + · · · + ( b m − a m ) x m = ( b 2 1 + · · · + b 2 m ) / 2 − ( a 2 1 + · · · + a 2 m ) / 2 . The closed half- space H ( a, b ) con taining a and with b oundary the bisector h yp erplane is the lo cus of a ll p oints suc h that ( b 1 − a 1 ) x 1 + · · · + ( b m − a m ) x m ≤ ( b 2 1 + · · · + b 2 m ) / 2 − ( a 2 1 + · · · + a 2 m ) / 2 , and the closed half-space H ( b, a ) con taining b and with b oundary the bisector hy p erplane is the lo cus o f all p oints suc h that ( b 1 − a 1 ) x 1 + · · · + ( b m − a m ) x m ≥ ( b 2 1 + · · · + b 2 m ) / 2 − ( a 2 1 + · · · + a 2 m ) / 2 . The closed half-space H ( a, b ) is the set o f all po in t s whose distance to a is less than or equal to the distance to b , and vice v ersa for H ( b, a ). Figure 8.2 sho ws the V oronoi diagram of a set of tw elv e p oin ts. 8.1. DIRICHLET–V ORONOI DIA GRAMS 141 Figure 8.2: A V o ro noi diagram In t he general case where E has dimension m , the deﬁnition o f the V oronoi diagram V or ( P ) of P is the same as D eﬁni tion 8 .1 , except that H ( p i , p j ) is the closed half-space con taining p i and ha ving the bisector hyperplane of a and b as boundary . Also, observ e that the con v ex hull o f P is a con v ex p olytop e. W e will no w state a lemma listing the main prop erties of V oronoi diagrams. It turns out that certain degenerate situations can b e a v oided if we assume that if P is a set of p oin ts in an aﬃne space of dimension m , then no m + 2 p oin ts from P b elong to the same ( m − 1)- sphere. W e will sa y that the p oin ts o f P are in gener a l p osition . Th us when m = 2, no 4 p oin ts in P are co cyclic, and when m = 3, no 5 p oin ts in P are on the same sphere. Lemma 8.1 Given a set P = { p 1 , . . . , p n } of n p oints in some Euclide an sp ac e E of dimen- sion m (say E m ), if the p oints in P ar e in gener al p osition and not in a c ommon hyp erplane then the V or onoi di a gr am of P satisﬁes the fol lowing c onditions: (1) Each r e gion V i is c onvex and c ontains p i in its in terior. (2) Each vertex of V i b elo ngs to m + 1 r e gions V j and to m + 1 e d g e s. (3) The r e gion V i is unb ounde d iﬀ p i b elo ngs to the b oundary of the c on v e x hul l of P . 142 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS (4) If p is a vertex that b e longs to the r e gions V 1 , . . . , V m +1 , then p i s the c enter of the ( m − 1) -s pher e S ( p ) determin e d by p 1 , . . . , p m +1 . F urthermor e, no p oi n t in P is ins ide the spher e S ( p ) (i.e., in the op en b al l asso ciate d with the spher e S ( p ) ). (5) If p j is a ne ar e st n eighb or of p i , then one of the fac es of V i is c ontaine d in the bise ctor hyp erpla ne of ( p i , p j ) . (6) n [ i =1 V i = E , and ◦ V i ∩ ◦ V j = ∅ , for al l i, j , with i 6 = j , wher e ◦ V i denotes the interior of V i . Pr o of . W e prov e only some of the statemen ts, lea ving the others as an exercise (or see Risler [31]). (1) Since V i = T j 6 = i H ( p i , p j ) and eac h half-space H ( p i , p j ) is con v ex, as an in tersection of con v ex sets, V i is con v ex. Also, since p i b elongs to t he interior of eac h H ( p i , p j ), the p oin t p i b elongs to the interior of V i . (2) Let F i,j denote V i ∩ V j . An y v ertex p of the V ononoi diagram of P mu st b elong to r faces F i,j . No w, giv en a v ector space E and an y t w o subspaces M and N of E , recall that w e hav e the Gr assmann r elation dim( M ) + dim( N ) = dim( M + N ) + dim ( M ∩ N ) . Then since p b elongs to the inters ection of the hyperplanes that form the b oundaries of the V i , and since a h yp erplane has dimens ion m − 1, by the Grassmann relation, w e mus t ha v e r ≥ m . F or simplicit y of notation, let us denote thes e faces b y F 1 , 2 , F 2 , 3 , . . . , F r,r +1 . Since F i,j = V i ∩ V j , w e ha v e F i,j = { p | d ( p, p i ) = d ( p, p j ) ≤ d ( p, p k ) , for all k 6 = i, j } , and since p ∈ F 1 , 2 ∩ F 2 , 3 ∩ · · · ∩ F r,r +1 , w e hav e d ( p, p 1 ) = · · · = d ( p, p r +1 ) < d ( p, p k ) for all k / ∈ { 1 , . . . , r + 1 } . This means that p is the cen ter of a sphere passing through p 1 , . . . , p r +1 and con taining no other p oin t in P . By the assumption that p oints in P a re in general p osition, w e m ust ha v e r ≤ m , and th us r = m . Th us, p b elongs to V 1 ∩ · · · ∩ V m +1 , but to no ot her V j with j / ∈ { 1 , . . . , m + 1 } . F urthermore, eve ry edge of the V oro noi diagram con taining p is the in tersection of m of the regions V 1 , . . . , V m +1 , and so there are m + 1 of them. F o r simplicit y , let us again consider the case where E is a plane. It should b e noted that certain V oronoi regions, although closed, ma y extend ve ry far. F igure 8.3 sho ws suc h an example. 8.1. DIRICHLET–V ORONOI DIA GRAMS 143 Figure 8.3: Another V oronoi diagram It is also p ossible for certain un b ounded regions to ha v e parallel edges. There are a num b er of methods for computing V oronoi diagrams. A fairly simple (a l- though not v ery eﬃcien t) metho d is to compute eac h V oronoi region V ( p i ) by interse cting the half-planes H ( p i , p j ). One wa y to do this is to construct successiv e con v ex p olygons that con v erge to the bo undary of the region. A t ev ery step w e in tersect the curren t con v ex p olygon with the bisector line of p i and p j . There are at most t w o inters ection p oints . W e also need a starting p olygon, and for this we can pic k a square con taining all the p oin ts. A naiv e implemen tation will run in O ( n 3 ). How eve r, the inters ection of half-planes can b e done in O ( n log n ), using the fact that the v ertices of a con ve x p olygon can be sorted. Th us, the ab o v e metho d runs in O ( n 2 log n ). Actually , there are faster methods (see Preparata and Shamos [30] or O’Rourke [29]), and it is p ossible to design algorithms running in O ( n log n ). The most direct metho d to obtain fa st algorithms is to use the “lifting metho d” discussed in Section 8.4, whereb y the original set of po in t s is lifted onto a parab oloid, a nd to use fast algorithms for ﬁnding a con v ex hull. A v ery inte resting (undirected) graph can b e o bta ined from the V oronoi diag ram as follo ws: The v ertices of this graph are the p o in t s p i (eac h corresponding to a unique region of V or ( P )), and there is an edge b et w een p i and p j iﬀ the regions V i and V j share an edge. The resulting g r a ph is called a D elaunay triangulation of the conv ex h ull of P , after Delaunay , who in v en ted this concep t in 1934. Suc h triangulations hav e remark able prop erties . 144 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Figure 8.4: Delaunay triangulation asso ciated with a V oronoi diagram Figure 8.4 sho ws the Delauna y triangulation a ss o ciated with the earlier V oronoi diagram of a set of t w elv e p oin ts. One has to b e careful to mak e sure that all the V oronoi v ertices ha v e b een computed b efore computing a Delauna y triangulation, since otherwise , some edges could b e miss ed. In Figure 8.5 illustrating suc h a situation, if the lo w est V oronoi ve rtex had not been computed (not sho wn on the diagra m!), the low est edge of the Delaunay triangulation would b e missing. The concept of a triangulatio n can b e generalized to dimension 3, or eve n to an y dimension m . 8.2 T riangulations The concept of a triangulation relies on the notion of pure simp licial complex deﬁned in Chapter 6. The reader should review Deﬁnition 6.2 and Deﬁnition 6.3. Deﬁnition 8.2 Giv en a subset, S ⊆ E m (where m ≥ 1), a triangulation of S is a pure (ﬁnite) simplicial complex, K , of dimension m suc h that S = | K | , that is, S is equal to the geometric realization of K . Giv en a ﬁnite set P of n p oints in the plane, and given a triangulation o f the conv ex h ull of P havin g P as its set of v ertices, observ e that the boundary of P is a con v ex p olygon. 8.2. TRIANGULA TIONS 145 Figure 8.5: Another Delaunay triangulation asso ciated with a V oronoi diagram Similarly , giv en a ﬁnite set P of p oin ts in 3-space, and g iv en a triang ulatio n of the conv ex h ull of P hav ing P as its set of v ertices, observ e that the b oundary of P is a conv ex p olyhedron. It is in teresting to kno w ho w many triangulations exist for a set of n p oin ts (in the plane or in 3-space), and it is a lso in teresting to kno w the n um b er of edges and faces in terms of the n um b er of v ertices in P . These questions can b e settled using the Euler–P oincar ´ e c haracteristic. W e say that a polygon in the plane is a simple p olygon iﬀ it is a connected closed p olygon suc h that no t w o edges inte rsect (excep t at a common ve rtex). Lemma 8.2 (1) F or any triangulation of a r e g i o n of the plane wh o se b oundary is a simple p olygon, letting v b e the numb er of vertic es, e the numb er of e dges, and f the numb er of triangles, we ha ve the “Euler f o rm ula” v − e + f = 1 . (2) F or any r e gion, S , in E 3 home omorphic to a close d b a l l and f or any triang ulation of S , letting v b e the numb er of vertic es, e the numb er of e dges, f the numb er of triangles, and t the numb e r of tetr ahe dr a, we have the “Euler formula” v − e + f − t = 1 . 146 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS (3) F urthermor e, f o r any triangulation of the c ombin atorial surfac e, B ( S ) , that is the b ounda ry of S , letting v ′ b e the numb er of vertic es, e ′ the numb er of e dges, and f ′ the numb e r of triangles, we have the “Euler formula” v ′ − e ′ + f ′ = 2 . Pr o of . All the statemen ts are immediate consequences o f Theore m 7.6. F or example, part (1) is obtained b y mapping the triangulation onto a sphere using in v erse stereographic pro- jection, sa y from the North p ole. Then , w e get a p olytop e on the sphere with an extra fa cet correspo nding to the “outside” o f the triangulation. W e ha v e to deduct this facet from the Euler c haracteristic of the p olytop e and this is wh y w e get 1 instead of 2. It is no w easy to see that in case (1), the n um b er of edges a nd faces is a linear function of the n um b er of v ertices and b oundary edges, and that in case (3), the n um b er of edges and faces is a linear function of the n um b er of v ertices. Indeed, in the case of a planar triangulation, each face has 3 edges, and if there are e b edges in the b oundary and e i edges not in the b oundary , eac h no nboundary edge is shared b y t w o faces, a nd th us 3 f = e b + 2 e i . Since v − e b − e i + f = 1, w e get v − e b − e i + e b / 3 + 2 e i / 3 = 1 , 2 e b / 3 + e i / 3 = v − 1 , and th us e i = 3 v − 3 − 2 e b . Since f = e b / 3 + 2 e i / 3, w e ha v e f = 2 v − 2 − e b . Similarly , since v ′ − e ′ + f ′ = 2 and 3 f ′ = 2 e ′ , w e easily get e = 3 v − 6 and f = 2 v − 4. Th us, giv en a set P o f n points, the n umber of triangles (and edges) for an y triangulation of the con v ex hull o f P using the n p oin ts in P for its v ertices is ﬁxed. Case (2) is tric kier, but it can b e sho wn that v − 3 ≤ t ≤ ( v − 1)( v − 2) / 2 . Th us, there can b e diﬀeren t n um b ers of tetrahedra fo r diﬀeren t triangulations of the conv ex h ull of P . Remark: The n um b ers of the form v − e + f and v − e + f − t are called Euler–Poinc ar ´ e char a cteristics . They are top ological in v ariants , in the sens e t hat they are the same for all triangulations of a giv en p olytop e. This is a fundamen tal fact of algebraic top ology . W e shall now in v estigate triangulations induced b y V oronoi diagrams. 8.3. DELA UNA Y TRIANGULA TIONS 147 Figure 8.6: A Delaunay triangulation 8.3 Delauna y T riangul atio ns Giv en a set P = { p 1 , . . . , p n } of n po in ts in the plane and the V oronoi diagram V or ( P ) for P , w e explaine d in Section 8.1 ho w to deﬁne an (undirected) g raph: The v ertices of this graph are the p oin ts p i (eac h corresponding to a unique region of V or ( P ) ) , and there is an edge b et w een p i and p j iﬀ the regions V i and V j share an edge. The resulting graph turns out to b e a triangulation of the conv ex hu ll of P havi ng P as its set of v ertices . Suc h a complex can b e deﬁned in general. F or an y set P = { p 1 , . . . , p n } of n p oin ts in E m , we sa y that a triangulation of the con v ex hull of P is asso ciate d with P if its set of v ertices is the set P . Deﬁnition 8.3 Let P = { p 1 , . . . , p n } b e a set of n p oin ts in E m , and let V or ( P ) b e the V oronoi diagram of P . W e deﬁne a complex D el ( P ) as follo ws. The complex D el ( P ) con tains the k -simplex { p 1 , . . . , p k +1 } iﬀ V 1 ∩ · · · ∩ V k +1 6 = ∅ , where 0 ≤ k ≤ m . The complex D el ( P ) is called t he D elaunay triangulation of the c onvex h ul l of P . Th us, { p i , p j } is an edge iﬀ V i ∩ V j 6 = ∅ , { p i , p j , p h } is a triangle iﬀ V i ∩ V j ∩ V h 6 = ∅ , { p i , p j , p h , p k } is a tetrahedron iﬀ V i ∩ V j ∩ V h ∩ V k 6 = ∅ , etc. F o r simplicit y , w e often write D el instead of D el ( P ). A Delauna y tria ngulatio n for a set of t w elv e po in t s is shown in Figure 8.6. Actually , it is not ob vious that D el ( P ) is a triangulation of the con v ex hull of P , but this can b e sho wn, as w ell as the prop erties listed in the follo wing lemma. 148 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Lemma 8.3 L et P = { p 1 , . . . , p n } b e a set of n p oints in E m , and assume that they ar e in gener al p osition. T hen the Delauna y triangulation of the c onvex hul l of P is inde e d a triangulation asso cia te d with P , and it satisﬁes the fol lowing pr op erties: (1) The b oundary of D el ( P ) is the c onvex hul l of P . (2) A triang ulation T ass o ciate d with P is the Dela unay triangulation D el ( P ) i ﬀ every ( m − 1) - s p her e S ( σ ) cir c umscrib e d ab out an m -simplex σ of T c ontains no other p oint fr om P (i . e ., the op en b al l asso ciate d with S ( σ ) c ontains no p oint fr om P ). The pro of can b e found in Risler [31] and O’Rourk e [29]. In t he case of a planar set P , it can also be show n that the Delaunay triangulation has the prop erty that it maximizes the minim um angle of the triangles in v olv ed in an y triangulation o f P . How ev er, this do es not c haracterize the Delauna y triangulatio n. G iv en a connected graph in the plane, it can a lso b e sho wn that any minimal spanning tree is contained in the Delauna y triangulation of the con v ex hull o f the set of vertic es o f the graph ( O ’R o urk e [29 ]). W e will no w explore brieﬂy the connecti on b etw een D elaunay triangulations and con v ex h ulls. 8.4 Delauna y T riangul atio ns and Con v ex Hulls In this section w e sho w that there is an in timate relationship b et w een con v ex h ulls a nd Delauna y triangulations. W e will see that giv en a set P of p oin ts in the Euclidean space E m of dimension m , w e can “lift” these p oin ts onto a parab oloid living in the space E m +1 of dimension m + 1, and that the D elauna y triangulation of P is the pro j ection of the down ward- facing faces of the conv ex h ull of the set of lifted p oin ts. This remark able connection w as ﬁrst disco v ered by Edelsbrunner and Seidel [16]. F or simplicit y , we consider the case of a set P of p oin ts in the plane E 2 , and w e assume that they are in general p osition. Consider the parab oloid of rev olution o f equation z = x 2 + y 2 . A p oin t p = ( x, y ) in the plane is lifted to the p o in t l ( p ) = ( X , Y , Z ) in E 3 , where X = x , Y = y , and Z = x 2 + y 2 . The ﬁrst crucial observ at io n is that a circle in the plane is lifted in to a plane curv e (an ellipse). Indeed, if suc h a circle C is deﬁned by the equation x 2 + y 2 + ax + by + c = 0 , since X = x , Y = y , and Z = x 2 + y 2 , b y eliminating x 2 + y 2 w e get Z = − ax − by − c, and th us X , Y , Z satisfy the linear equation aX + bY + Z + c = 0 , 8.4. DELA UNA Y TRIANGULA TIONS AND CONVEX HULLS 149 0 2.5 5 7.5 10 x 0 2.5 5 7.5 10 y 0 5 10 z 0 2.5 5 7.5 10 x 0 2.5 5 7.5 10 y 0 5 10 z Figure 8.7: A Delaunay triangulation and its lifting to a parab oloid whic h is the equation of a plane. Th us, the interse ction of the cylinder of revolution consisting of the lines parallel to the z -axis and passing through a p oin t of the circle C with the parab oloid z = x 2 + y 2 is a planar curv e (an ellipse). W e can compute the con v ex h ull of the set of lifted p oin ts. Let us fo cus on the dow nw ard- facing faces of this conv ex h ull. Let ( l ( p 1 ) , l ( p 2 ) , l ( p 3 )) b e suc h a face. The p oin ts p 1 , p 2 , p 3 b elong to the set P . W e claim that no other p oin t from P is inside t he circle C . Indeed, a p oin t p inside the circle C w ould lift to a p oin t l ( p ) on the para boloid. Since no four p oin ts are co cyclic , one of the four p oin ts p 1 , p 2 , p 3 , p is further from O than the others; sa y this p oin t is p 3 . Then, the fa ce ( l ( p 1 ) , l ( p 2 ) , l ( p )) would b e b elo w the face ( l ( p 1 ) , l ( p 2 ) , l ( p 3 )), con tradicting the fact that ( l ( p 1 ) , l ( p 2 ) , l ( p 3 )) is one o f the down ward-facing f a ces o f the con v ex hul l of P . But then, b y prop ert y (2) of Lemma 8.3, the triangle ( p 1 , p 2 , p 3 ) w ould b elong to the Delauna y triangulation of P . Therefore, w e hav e sho wn that the pr oje ction of the p art of the c onvex hul l of the lifte d set l ( P ) c o n sisting of the downwar d - facing fac es is the Del a unay triangulation of P . Figure 8.7 sho ws the lifting of the Delauna y triangulation sho wn earlier. Another example of the lifting of a Delauna y triangulation is sho wn in Figure 8.8. The fact that a Delauna y triangulation can b e obtained b y pro jecting a low er con v ex h ull can b e used to ﬁnd eﬃcien t alg o rithms for computing a D elaun ay triangulation. It also holds for higher dimens ions. The V oronoi diagram itself can also b e obtained from the lifted set l ( P ). Ho w ev er, this time, w e need to consider tangen t planes to the parab oloid at the lifted p oin ts. It is fairly 150 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS 0 2.5 5 7.5 10 x 0 2.5 5 7.5 10 y 0 5 10 z 0 2.5 5 7.5 10 x 0 2.5 5 7.5 10 y 0 5 10 z Figure 8.8: Another Delaunay triangulation and its lifting to a parab oloid ob vious that the tangen t plane at the lift ed p oint ( a, b, a 2 + b 2 ) is z = 2 ax + 2 by − ( a 2 + b 2 ) . Giv en t w o distinct lifted p oin ts ( a 1 , b 1 , a 2 1 + b 2 1 ) and ( a 2 , b 2 , a 2 2 + b 2 2 ), the in tersection of the tangen t planes at these p oin ts is a line b elonging to the plane of equation ( b 1 − a 1 ) x + ( b 2 − a 2 ) y = ( b 2 1 + b 2 2 ) / 2 − ( a 2 1 + a 2 2 ) / 2 . No w, if w e pro j ect this plane onto the xy -plane, w e see that the ab ov e is precisely the equation of the bisector line of the t w o p oin ts ( a 1 , b 1 ) and ( a 2 , b 2 ). Therefore, if we lo ok at the p ar ab olo i d fr om z = + ∞ (with the p ar ab oloid tr ansp ar ent), the pr oje ction of the tangent planes at the lifte d p oints is the V or onoi diagr a m ! It should b e noted that the “dualit y” b et w een the Delauna y triangulation, whic h is the pro jection of the con v ex h ull of the lifted set l ( P ) vie we d fro m z = −∞ , and the V oronoi diagram, whic h is the pro jection of the tangen t planes at the lifted set l ( P ) view ed from z = + ∞ , is reminiscen t of the p olar dualit y with respect to a quadric. This duality will b e thoroughly in v estigated in Section 8.6. The reader in terested in alg orithms for ﬁnding V oronoi diagrams and Delauna y triangu- lations is r eferred to O’Rourk e [29], Preparata and Shamos [30], Boissonnat and Yvinec [8], de Berg, V an Krev eld, Overm ars, and Sc h w arzk opf [5], and Risler [31]. 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 151 8.5 Stereograph ic Pro jection and the Space o f Generalized Sphe r e s Bro wn app ears to b e the ﬁrst p erson who observ ed that V oronoi diagrams and conv ex hulls are related via in v ersion with resp ect to a sphere [11]. In fact, more generally , it turns out that V oronoi diagrams, Delauna y T riangulations and their prop erties can also b e nicely explain ed using stereographic pro jection and its in v erse, although a rigorous justiﬁcation of wh y this “w orks” is not as simple as it migh t app ear. The adv antage of stere ogra phic pro jection ov er the lifting onto a para boloid is that the ( d -)sphere is compact. Since the stereographic pro jection and its in v erse map ( d − 1)-spheres to ( d − 1)-spheres (or hy p erplanes), all the crucial prop erties of Delauna y triangulations are preserv ed. The purp ose of this section is to establis h the prop erties of stereographic pro jection (and its inv erse) that will b e needed in Section 8.6. Recall that the d -spher e , S d ⊆ E d +1 , is giv en b y S d = { ( x 1 , . . . , x d +1 ) ∈ E d +1 | x 2 1 + · · · + x 2 d + x 2 d +1 = 1 } . It will b e con v enien t to write a p oin t, ( x 1 , . . . , x d +1 ) ∈ E d +1 , as z = ( x, x d +1 ), with x = ( x 1 , . . . , x d ). W e denote N = (0 , . . . , 0 , 1) (with d zeros) as ( 0 , 1) and call it the north p ol e and S = (0 , . . . , 0 , − 1) (with d zeros) as ( 0 , − 1) and call it the south p ole . W e also write k z k = ( x 2 1 + · · · + x 2 d +1 ) 1 2 = ( k x k 2 + x 2 d +1 ) 1 2 (with k x k = ( x 2 1 + · · · + x 2 d ) 1 2 ). With these notations, S d = { ( x, x d +1 ) ∈ E d +1 | k x k 2 + x 2 d +1 = 1 } . The ster e o gr aphic pr oje ction fr om the north p ole, σ N : ( S d − { N } ) → E d , is the restriction to S d of the cen tral pro jection from N onto the hyperplane, H d +1 (0) ∼ = E d , o f equation x d +1 = 0; that is, M 7→ σ N ( M ) where σ N ( M ) is the in tersection of the line, h N , M i , through N and M with H d +1 (0). Since the line through N and M = ( x, x d +1 ) is giv en parametrically b y h N , M i = { (1 − λ )( 0 , 1) + λ ( x, x d +1 ) | λ ∈ R } , the in tersection, σ N ( M ), of this line with the hy p erplane x d +1 = 0 corresp onds to the v alue of λ suc h that (1 − λ ) + λx d +1 = 0 , that is, λ = 1 1 − x d +1 . Therefore, the co ordinates of σ N ( M ), with M = ( x, x d +1 ), are giv en b y σ N ( x, x d +1 ) =  x 1 − x d +1 , 0  . 152 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Let us ﬁnd the in v erse, τ N = σ − 1 N ( P ), o f an y P ∈ H d +1 (0). This time, τ N ( P ) is the in ter- section of the line, h N , P i , through P ∈ H d +1 (0) and N with the sphere , S d . Since the line through N and P = ( x, 0) is giv en parametrically by h N , P i = { (1 − λ )( 0 , 1) + λ ( x, 0 ) | λ ∈ R } , the interse ction, τ N ( P ), of this line with the sphere S d correspo nds to the nonzero v alue of λ suc h that λ 2 k x k 2 + (1 − λ ) 2 = 1 , that is λ ( λ ( k x k 2 + 1) − 2 ) = 0 . Th us, w e g et λ = 2 k x k 2 + 1 , from whic h w e get τ N ( x, 0) =  2 x k x k 2 + 1 , 1 − 2 k x k 2 + 1  = 2 x k x k 2 + 1 , k x k 2 − 1 k x k 2 + 1 ! . W e lea v e it as an exercise to the reader to v erify that τ N ◦ σ N = id and σ N ◦ τ N = id. W e can also deﬁne the ster e o gr ap hic pr o je ction fr om the south p ole, σ S : ( S d − { S } ) → E d , and its in v erse, τ S . Again, the computations are left as a simple exercis e to the reader. The ab o v e computations are summarized in the followi ng deﬁnition: Deﬁnition 8.4 The ster e o gr aphic pr oje ction f r om the north p ole , σ N : ( S d − { N } ) → E d , is the map giv en b y σ N ( x, x d +1 ) =  x 1 − x d +1 , 0  ( x d +1 6 = 1) . The in v erse of σ N , denoted τ N and called inverse ster e o gr aphic pr oje ction fr om the north p ole is giv en b y τ N ( x, 0) = 2 x k x k 2 + 1 , k x k 2 − 1 k x k 2 + 1 ! . Remark: An inversion of c enter C and p ower ρ > 0 is a g eometric transformation, f : ( E d +1 − { C } ) → E d +1 , deﬁned so that for any M 6 = C , the p oin ts C , M and f ( M ) are collinear and k CM kk Cf ( M ) k = ρ. 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 153 Equiv alen tly , f ( M ) is give n b y f ( M ) = C + ρ k CM k 2 CM . Clearly , f ◦ f = id on E d +1 − { C } , so f is in v ertible and the reader will che ck that if w e pic k t he cen ter of in v ersion to b e the north p ole and if w e set ρ = 2 , then t he co ordinates of f ( M ) are giv en by y i = 2 x i x 2 1 + · · · + x 2 d + x 2 d +1 − 2 x d +1 + 1 1 ≤ i ≤ d y d +1 = x 2 1 + · · · + x 2 d + x 2 d +1 − 1 x 2 1 + · · · + x 2 d + x 2 d +1 − 2 x d +1 + 1 , where ( x 1 , . . . , x d +1 ) are the co ordinates of M . In particular, if w e restrict o ur in v ersion to the unit sphere, S d , as x 2 1 + · · · + x 2 d + x 2 d +1 = 1, w e get y i = x i 1 − x d +1 1 ≤ i ≤ d y d +1 = 0 , whic h means that o ur inv ersion restricted to S d is simply the stereographic pro jection, σ N (and the in v erse of our in v ersion restricted to the h yp erplane, x d +1 = 0, is the in v erse stereographic pro jection, τ N ). W e will no w sho w that the image of any ( d − 1 ) - sphe re, S , on S d not passing through the north p ole, t hat is, the inters ection, S = S d ∩ H , of S d with any hyperplane, H , not passing through N is a ( d − 1)-sphere. Here, w e are assuming that S has p ositiv e radius, that is, H is not tangen t to S d . Assume that H is giv en by a 1 x 1 + · · · + a d x d + a d +1 x d +1 + b = 0 . Since N / ∈ H , w e mu st ha v e a d +1 + b 6 = 0. F or an y ( x, x d +1 ) ∈ S d , write σ N ( x, x d +1 ) = ( X, 0). Since X = x 1 − x d +1 , w e get x = X (1 − x d +1 ) and using the fact that ( x, x d +1 ) also b elongs to H we will express x d +1 in terms of X and then ﬁnd an equation for X whic h will sho w that X b elongs to a ( d − 1 )-sphere . Indeed, ( x, x d +1 ) ∈ H implies that d X i =1 a i X i (1 − x d +1 ) + a d +1 x d +1 + b = 0 , 154 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS that is, d X i =1 a i X i + ( a d +1 − d X j =1 a j X j ) x d +1 + b = 0 . If P d j =1 a j X j = a d +1 , then a d +1 + b = 0, whic h is imp ossible. Therefore, w e get x d +1 = − b − P d i =1 a i X i a d +1 − P d i =1 a i X i and so, 1 − x d +1 = a d +1 + b a d +1 − P d i =1 a i X i . Plugging x = X (1 − x d +1 ) in the equation, k x k 2 + x d d +1 = 1, of S d , w e get (1 − x d +1 ) 2 k X k 2 + x 2 d +1 = 1 , and replacing x d +1 and 1 − x d +1 b y their expression in terms of X , w e get ( a d +1 + b ) 2 k X k 2 + ( − b − d X i =1 a i X i ) 2 = ( a d +1 − d X i =1 a i X i ) 2 that is, ( a d +1 + b ) 2 k X k 2 = ( a d +1 − d X i =1 a i X i ) 2 − ( b + d X i =1 a i X i ) 2 = ( a d +1 + b )( a d +1 − b − 2 d X i =1 a i X i ) 2 whic h yields ( a d +1 + b ) 2 k X k 2 + 2( a d +1 + b )( d X i =1 a i X i ) = ( a d +1 + b )( a d +1 − b ) , that is, k X k 2 + 2 d X i =1 a i a d +1 + b X i − a d +1 − b a d +1 + b = 0 , whic h is indeed the equation of a ( d − 1)-sphere in E d . Therefore, when N / ∈ H , the image of S = S d ∩ H by σ N is a ( d − 1)-sphere in H d +1 (0) = E d . 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 155 If the h yp erplane, H , contains the north p ole, then a d +1 + b = 0, in whic h case, for ev ery ( x, x d +1 ) ∈ S d ∩ H , w e ha v e d X i =1 a i x i + a d +1 x d +1 − a d +1 = 0 , that is, d X i =1 a i x i − a d +1 (1 − x d +1 ) = 0 , and except for the north p o le, we ha v e d X i =1 a i x i 1 − x d +1 − a d +1 = 0 , whic h sho ws that d X i =1 a i X i − a d +1 = 0 , the in tersec tion of the hyperplanes H and H d +1 (0) Therefore, the image of S d ∩ H by σ N is the h yp erplane in E d whic h is the in tersection of H with H d +1 (0). W e will also pro v e that τ N maps ( d − 1)-spheres in H d +1 (0) t o ( d − 1)-spheres o n S d not passing through the north p ole. Assume that X ∈ E d b elongs to the ( d − 1)-sphere of equation d X i =1 X 2 i + d X j =1 a j X j + b = 0 . F o r any ( X , 0) ∈ H d +1 (0), w e kno w that ( x, x d +1 ) = τ N ( X , 0) is giv en by ( x, x d +1 ) = 2 X k X k 2 + 1 , k X k 2 − 1 k X k 2 + 1 ! . Using the equation of the ( d − 1 )-sphere , we get x = 2 X − b + 1 − P d j =1 a j X j and x d +1 = − b − 1 − P d j =1 a j X j − b + 1 − P d j =1 a j X j . Then, w e get d X i =1 a i x i = 2 P d j =1 a j X j − b + 1 − P d j =1 a j X j , 156 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS whic h yields ( − b + 1)( d X i =1 a i x i ) − ( d X i =1 a i x i )( d X j =1 a j X j ) = 2 d X j =1 a j X j . F ro m the ab o v e, w e get d X i =1 a i X i = ( − b + 1)( P d i =1 a i x i ) P d i =1 a i x i + 2 . Plugging this express ion in the form ula for x d +1 ab o v e, w e get x d +1 = − b − 1 − P d i =1 a i x i − b + 1 , whic h yields d X i =1 a i x i + ( − b + 1) x d +1 + ( b + 1 ) = 0 , the equation of a h yp erplane, H , not passing through the north p ole. Therefore, the image of a ( d − 1)-sphere in H d +1 (0) is inde ed the inte rsection, H ∩ S d , of S d with a hyperplane not passing through N , that is, a ( d − 1)-sphere on S d . Giv en any hy p erplane, H ′ , in H d +1 (0) = E d , sa y of equation d X i =1 a i X i + b = 0 , the image of H ′ under τ N is a ( d − 1 ) - sphe re on S d , the in tersection of S d with the h yperplane, H , passing t hroug h N and determined as follo ws: F or an y ( X , 0) ∈ H d +1 (0), if τ N ( X , 0) = ( x, x d +1 ), then X = x 1 − x d +1 and so, ( x, x d +1 ) satisﬁes the equation d X i =1 a i x i + b (1 − x d +1 ) = 0 , that is, d X i =1 a i x i − bx d +1 + b = 0 , whic h is indeed the equation o f a hyperplane, H , passing through N . W e summarize all this in the follo wing prop osition: 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 157 Prop osition 8.4 The ster e o g r aphic pr oje ction, σ N : ( S d − { N } ) → E d , induc es a bije ction, σ N , b etwe en the set o f ( d − 1) - spher es on S d and the union of the set of ( d − 1) -sph er es in E d with the set of hyp erplanes in E d ; every ( d − 1) -spher e on S d not p a s sing thr ough the north p ol e is mapp e d to a ( d − 1) -spher e in E d and every ( d − 1) -sp h er e on S d p as s ing thr ough the north p ole is mapp e d to a hyp erplane in E d . In fact, σ N maps the hyp erplane a 1 x 1 + · · · + a d x d + a d +1 x d +1 + b = 0 not p assing thr ough the north p ole ( a d +1 + b 6 = 0 ) to the ( d − 1) -s pher e d X i =1 X 2 i + 2 d X i =1 a i a d +1 + b X i − a d +1 − b a d +1 + b = 0 and the hyp erplane d X i =1 a i x i + a d +1 x d +1 − a d +1 = 0 thr ough the north p ole to the hyp erplane d X i =1 a i X i − a d +1 = 0; the map τ N = σ − 1 N maps the ( d − 1) -sph e r e d X i =1 X 2 i + d X j =1 a j X j + b = 0 to the hyp erplane d X i =1 a i x i + ( − b + 1) x d +1 + ( b + 1) = 0 not p assing thr ough the north p ole and the hyp erplane d X i =1 a i X i + b = 0 to the hyp erplane d X i =1 a i x i − bx d +1 + b = 0 thr ough the north p ole. 158 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Prop osition 8.4 raises a natural question: What do the hyperplanes, H , in E d +1 that do not in tersect S d correspo nd to , if they corresp ond to an ything at a ll? The ﬁrst thing to observ e is that the geometric deﬁnition of the stereographic pro jection and its in v erse mak e it clear that the h yp erplanes associated with ( d − 1 )-sphere s in E d (b y τ N ) do in tersect S d . Now, when we write the equation of a ( d − 1)- sphe re, S , sa y d X i =1 X 2 i + d X i =1 a i X i + b = 0 w e are implicitly assuming a condition on the a i ’s and b that ensures that S is not the empt y sphere, that is, that its radius, R , is p ositiv e (or zero). By “completing the square”, the ab o v e equation can b e rewritten as d X i =1  X i + a i 2  2 = 1 4 d X i =1 a 2 i − b, and so the radius, R , of our sphere is giv en by R 2 = 1 4 d X i =1 a 2 i − b whereas its cen ter is the p oin t, c = − 1 2 ( a 1 , . . . , a d ). Th us, our sphe re is a “real” sphere of p ositiv e radius iﬀ d X i =1 a 2 i > 4 b or a single p oin t, c = − 1 2 ( a 1 , . . . , a d ), iﬀ P d i =1 a 2 i = 4 b . What happ ens when d X i =1 a 2 i < 4 b ? In this case, if w e allow “complex p oin ts”, that is, if w e consider solutions of our equation d X i =1 X 2 i + d X i =1 a i X i + b = 0 o v er C d , then w e get a “complex” sphere of (pure) imaginary radius, i 2 q 4 b − P d i =1 a 2 i . The funn y thing is t hat our computations carry o v er unc hanged and the image of the complex sphere, S , is still the h yp erplane, H , giv en d X i =1 a i x i + ( − b + 1) x d +1 + ( b + 1 ) = 0 . 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 159 Ho w ev er, this time, ev en though H do es not hav e an y “real” inters ection p oin ts with S d , we can sho w that it do es in tersect the “complex sphere”, S d = { ( z 1 , . . . , z d +1 ) ∈ C d +1 | z 2 1 + · · · + z 2 d +1 = 1 } in a nonempt y set of p oin ts in C d +1 . It follow s from all this that σ N and τ N establish a bijection b et w een the set of all h y- p erplanes in E d +1 min us the h yp erplane , H d +1 (of equation x d +1 = 1), tangent to S d at the north p ole, with the union of four sets: (1) The set o f all (real) ( d − 1)-spheres of p ositiv e radius; (2) The set o f all (complex) ( d − 1)- sph eres of imaginary radius; (3) The set o f all h yp erplanes in E d ; (4) The set o f all p oin ts of E d (view ed as spheres of radius 0). Moreo v er, set (1) corresponds to the h yp erplanes that inter sect the in terior of S d and do not pass through the north p ole; set (2) corresp onds to the h yp erplanes that do not inters ect S d ; set (3) corresponds to the h yperplanes that pass through the north p ole min us the tangent h yp erplane at the north p ole; and set (4) corresp onds to the h yp erplanes that are tangent to S d , min us the tangen t hyperplane at the north p ole. It is conv enien t to add the “p oin t at inﬁnit y”, ∞ , to E d , b ecause then the ab o v e bijection can b e extended to map the tangent hyperplane at the north p ole to ∞ . The union of t hese four sets (with ∞ a dded) is called the set of gene r alize d sph er es , sometime s, denoted S ( E d ). This is a fairly complicated space. F or one thing, top ologically , S ( E d ) is homeomorphic to the pro jectiv e space P d +1 with one p oin t remo v ed (the p oin t corresponding to the “hyperplane at inﬁnit y”), and this is not a simple space. W e can get a sligh tly more concrete “‘picture” of S ( E d ) b y lo oking at the p olars of the hyperplanes w.r.t. S d . Then , the “ r eal” spheres correspo nd to the p oin ts strictly outside S d whic h do not b elong to the tangen t hyperplane at the norh p ole; the complex spheres correspond to the p oin ts in the in terior of S d ; the p oin ts of E d ∪ {∞} corresp ond to the p oin ts on S d ; the h yp erplane s in E d correspo nd to the p oin ts in the tangen t h yp erplane at the norh p ole expect for the north p ole. Unfortunately , the p oles of h yp erplanes through the origin are undeﬁned. This can b e ﬁxed by em b eddin g E d +1 in its pro jectiv e completion, P d +1 , but w e will not go in to this. There ar e other w a ys of dealing rigorously with the set of g en eralized spheres. One metho d desc rib ed b y Boissonnat [8] is to use the em bedding where the sphere, S , o f equation d X i =1 X 2 i − 2 d X i =1 a i X i + b = 0 is mapp ed to the p o in t ϕ ( S ) = ( a 1 , . . . , a d , b ) ∈ E d +1 . 160 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS No w, b y a previous computation we kno w that b = d X i =1 a 2 i − R 2 , where c = ( a 1 , . . . , a d ) is the cen ter of S and R is its radius. The quan tit y P d i =1 a 2 i − R 2 is kno wn as the p ower of the origin w.r.t. S . In g eneral, the p ower of a p oin t, X ∈ E d , is deﬁned as ρ ( X ) = k cX k 2 − R 2 , whic h, after a momen t of thought, is just ρ ( X ) = d X i =1 X 2 i − 2 d X i =1 a i X i + b. No w, since p oin ts correspond to spheres of radius 0, we see that the image of the p oin t, X = ( X 1 , . . . , X d ), is l ( X ) = ( X 1 , . . . , X d , d X i =1 X 2 i ) . Th us, in this mo del, p oin ts of E d are lifted to the h yp erbolo id, P ⊆ E d +1 , of equation x d +1 = d X i =1 x 2 i . Actually , this metho d do es not deal with h yp erplanes but it is p ossible t o do so. The tric k is to consider equations of a sligh tly more general form that capture b oth spheres and h yp erplanes , namely , equations of the form c d X i =1 X 2 i + d X i =1 a i X i + b = 0 . Indeed, when c = 0, w e do get a h yp erplane! Now , to carry out this metho d w e really need to conside r equations up to a nonzero scalars, that is, w e consider the pro jectiv e space, P ( b S ( E d )), associated with the v ector space, b S ( E d ), consisting of the ab o v e equations . Then, it turns out that the quan tit y  ( a, b, c ) = 1 4 ( d X i =1 a 2 i − 4 bc ) (with a = ( a 1 , . . . , a d )) deﬁnes a quadratic form on b S ( E d ) whose correspo nding bilinear form, ρ (( a, b, c ) , ( a ′ , b ′ , c ′ )) = 1 4 ( d X i =1 a i a ′ i − 2 bc ′ − 2 b ′ c ) , 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 161 has a natural interpre tation (with a = ( a 1 , . . . , a d ) and a ′ = ( a ′ 1 , . . . , a ′ d )). Indeed, orthogo- nalit y with resp ect to ρ (that is, when ρ (( a, b, c ) , ( a ′ , b ′ , c ′ )) = 0) sa ys that the corresp onding spheres deﬁned by ( a, b, c ) and ( a ′ , b ′ , c ′ ) are orthogonal, that the corresp onding hyperplanes deﬁned b y ( a, b, 0) and ( a ′ , b ′ , 0) are orthogonal, etc. T he reader who w an ts to read more ab out this a pproa ch should consult Berger (V olume I I) [6]. There is a simple relationship b et w een the lifting on to a hyperb oloid and the lifting on to S d using the in v erse stereographic pro jection map because the sphere a nd the parab oloid are pro jectiv ely equiv alen t, a s w e show ed for S 2 in Section 5.1. Recall that the h yp erb oloid, P , in E d +1 is giv en b y the equation x d +1 = d X i =1 x 2 i and of course, the sphere S d is giv en b y d +1 X i =1 x 2 i = 1 . Consider the “pro jectiv e transformation”, Θ, of E d +1 giv en b y z i = x i 1 − x d +1 , 1 ≤ i ≤ d z d +1 = x d +1 + 1 1 − x d +1 . Observ e that Θ is undeﬁned on the h yp erplane, H d +1 , tangen t to S d at the north p ole and that its ﬁrst d comp onen t are iden tical to those of the stereographic pro jection! Then, w e immediately ﬁnd that x i = 2 z i 1 + z d +1 , 1 ≤ i ≤ d x d +1 = z d +1 − 1 1 + z d +1 . Conseque ntly , Θ is a bijection b et w een E d +1 − H d +1 and E d +1 − H d +1 ( − 1), where H d +1 ( − 1) is the h yp erplane of equation x d +1 = − 1. The fact that Θ is undeﬁned on the h yperplane, H d +1 is not a problem as far as mapping the sphere to the parab oloid b ecaus e the north p ole is the only p oin t that do es hav e not an image. Ho w ev er, later on when we cons ider the V o ronoi polyhedron, V ( P ), of a lifted set of p oin ts, P , w e will hav e more serious problems because in general, suc h a po lyhe dron in tersects b oth h yp erplanes H d +1 and H d +1 ( − 1). This means that Θ will not b e w ell-deﬁned on the whole of V ( P ) nor will it b e surjectiv e on its image. T o remedy this diﬃcu lty , w e 162 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS will work with pro jectiv e completions. Basically , this amoun ts to ch asing denominators and homogenizing equations but w e also ha v e to b e careful in dealing with con v exit y and this is where the pro jectiv e po lyhedra (studied in Section 5 .2 ) will come handy . So, let us consider the pro jectiv e sphere , S d ⊆ P d +1 , giv en b y the equation d +1 X i =1 x 2 i = x 2 d +2 and the parab oloid, P ⊆ P d +1 , giv en b y the equation x d +1 x d +2 = d X i =1 x 2 i . Let θ : P d +1 → P d +1 b e the pro jectivit y induced b y the linear map, b θ : R d +2 → R d +2 , giv en b y z i = x i , 1 ≤ i ≤ d z d +1 = x d +1 + x d +2 z d +2 = x d +2 − x d +1 , whose in v erse is g iven b y x i = z i , 1 ≤ i ≤ d x d +1 = z d +1 − z d +2 2 x d +2 = z d +1 + z d +2 2 . If w e plug these formul ae in the equation of S d , w e get 4( d X i =1 z 2 i ) + ( z d +1 − z d +2 ) 2 = ( z d +1 + z d +2 ) 2 , whic h simpliﬁes to z d +1 z d +2 = d X i =1 z 2 i . Therefore, θ ( S d ) = P , that is, θ maps the sphere to the h yp erboloid. Observ e that the north p ole, N = (0 : · · · : 0 : 1 : 1), is mapp ed to the p oin t at inﬁnit y , (0 : · · · : 0 : 1 : 0). The map Θ is the restriction o f θ to the aﬃne patc h, U d +1 , and as suc h, it can b e fruitfully described as the comp osition of b θ with a suitable pro jection on to E d +1 . F or this, a s w e ha v e 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 163 done b efore, w e iden tify E d +1 with the h yp erplane, H d +2 ⊆ E d +2 , of equation x d +2 = 1 (using the injection, i d +2 : E d +1 → E d +2 , where i j : E d +1 → E d +2 is the injection giv en b y ( x 1 , . . . , x d +1 ) 7→ ( x 1 , . . . , x j − 1 , 1 , x j +1 , . . . , x d +1 ) for an y ( x 1 , . . . , x d +1 ) ∈ E d +1 ). F or eac h i , with 1 ≤ i ≤ d + 2, let π i : ( E d +2 − H i (0)) → E d +1 b e t he pro jection of cen ter 0 ∈ E d +2 on to the h yp erplane, H i ⊆ E d +2 , of equation x i = 1 ( H i ∼ = E d +1 and H i (0) ⊆ E d +2 is the h yp erplane of equation x i = 0) give n b y π i ( x 1 , . . . , x d +2 ) =  x 1 x i , . . . , x i − 1 x i , x i +1 x i , . . . , x d +2 x i  ( x i 6 = 0) . Geometrically , for an y x / ∈ H i (0), the image, π i ( x ), of x is the in tersection of the line through the origin and x with the h yp erplane, H i ⊆ E d +2 of equation x i = 1. Observ e that the map, π i : ( E d +2 − H d +2 (0)) → E d +1 , is an “aﬃne” v ersion of the bijection, ϕ i : U i → R d +1 , of Section 5.1. Then, w e hav e Θ = π d +2 ◦ b θ ◦ i d +2 . If w e iden tify H d +2 and E d +1 , w e ma y write with a sligh t abuse o f notation, Θ = π d +2 ◦ b θ . Besides θ , we need to deﬁne a few more maps in order to establish the connection b et w een the Delauna y complex on S d and the Delauna y complex o n P . W e use the con v en tion of denoting the extension to pro jectiv e space s of a map, f , deﬁned betw een Euclidean spaces, b y e f . The Euclidean orthogonal pro j ecti on, p i : R d +1 → R d , is giv en b y p i ( x 1 , . . . , x d +1 ) = ( x 1 , . . . , x i − 1 , x i +1 , . . . , x d +1 ) and e p i : P d +1 → P d denotes the pro jection from P d +1 on to P d giv en by e p i ( x 1 : · · · : x d +2 ) = ( x 1 : · · · : x i − 1 : x i +1 : · · · : x d +2 ) , whic h is undeﬁne d at the p oint (0 : · · · : 1 : 0 : · · · : 0), where the “1” is in the i th slot. The map e π N : ( P d +1 − { N } ) → P d is the central pro jection from the north p ole on to P d giv en b y e π N ( x 1 : · · · : x d +1 : x d +2 ) = ( x 1 : · · · : x d : x d +2 − x d +1 ) . A geometric inte rpretation of e π N will b e needed later in certain pro ofs. If w e iden tify P d with t he h yp erplane, H d +1 (0) ⊆ P d +1 , of equation x d +1 = 0, then w e claim that for any , x 6 = N , the p oin t e π N ( x ) is the in tersection of the line through N and x with the h yp erplane, H d +1 (0). Indeed, parametrically , the line, h N , x i , through N = (0 : · · · : 0 : 1 : 1) and x is giv en by h N , x i = { ( µx 1 : · · · : µx d : λ + µx d +1 : λ + µx d +2 ) | λ, µ ∈ R , λ 6 = 0 or µ 6 = 0 } . 164 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS The line h N , x i inte rsects the hyperplane x d +1 = 0 iﬀ λ + µx d +1 = 0 , so w e can pic k λ = − x d +1 and µ = 1, whic h yields the in tersection p oin t, ( x 1 : · · · : x d : 0 : x d +2 − x d +1 ) , as claimed. W e also hav e the pro jectiv e v ersions of σ N and τ N , denoted e σ N : ( S d − { N } ) → P d and e τ N : P d → S d ⊆ P d +1 , giv en b y: e σ N ( x 1 : · · · : x d +2 ) = ( x 1 : · · · : x d : x d +2 − x d +1 ) and e τ N ( x 1 : · · · : x d +1 ) = 2 x 1 x d +1 : · · · : 2 x d x d +1 : d X i =1 x 2 i − x 2 d +1 : d X i =1 x 2 i + x 2 d +1 ! . It is a n easy exercise to che ck that the image of S d − { N } b y e σ N is U d +1 and that e σ N and e τ N ↾ U d +1 are m utual inv erses. Observ e t hat e σ N = e π N ↾ S d , the restriction of t he pro jection, e π N , to the sphe re, S d . The lifting, e l : E d → P ⊆ P d +1 , is giv en b y e l ( x 1 , . . . , x d ) = x 1 : · · · : x d : d X i =1 x 2 i : 1 ! and the em b edding, ψ d +1 : E d → P d , (the map ψ d +1 deﬁned in Section 5.1) is giv en by ψ d +1 ( x 1 , . . . , x d ) = ( x 1 : · · · : x d : 1) . Then, w e easily c hec k Prop osition 8.5 The maps, θ , e π N , e τ N , e p d +1 , e l and ψ d +1 deﬁne d b efor e satisfy the e quations e l = θ ◦ e τ N ◦ ψ d +1 e π N = e p d +1 ◦ θ e τ N ◦ ψ d +1 = ψ d +2 ◦ τ N e l = ψ d +2 ◦ l l = Θ ◦ τ N . Pr o of . Let us c hec k the ﬁrst equation lea ving the others as an exercise. R ecall that θ is giv en by θ ( x 1 : · · · : x d +2 ) = ( x 1 : · · · : x d : x d +1 + x d +2 : x d +2 − x d +1 ) . 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 165 Then, as e τ N ◦ ψ d +1 ( x 1 , . . . , x d ) = 2 x 1 : · · · : 2 x d : d X i =1 x 2 i − 1 : d X i =1 x 2 i + 1 ! , w e get θ ◦ e τ N ◦ ψ d +1 ( x 1 , . . . , x d ) = 2 x 1 : · · · : 2 x d : 2 d X i =1 x 2 i : 2 ! = x 1 : · · · : x d : d X i =1 x 2 i : 1 ! = e l ( x 1 , . . . , x d ) , as claimed. W e will also need some prop erties of the pro jection π d +2 and of Θ and for this, let H d + = { ( x 1 , . . . , x d ) ∈ E d | x d > 0 } a nd H d − = { ( x 1 , . . . , x d ) ∈ E d | x d < 0 } . Prop osition 8.6 The pr oje ction, π d +2 , has the fol lowing pr op erties: (1) F or every hyp erplane, H , thr ough the origin, π d +2 ( H ) is a hyp erplane in H d +2 . (2) Given any set of p oints, { a 1 , . . . , a n } ⊆ E d +2 , if { a 1 , . . . , a n } is c ontaine d in the op en half-sp ac e ab ove the hyp e rplane x d +2 = 0 or { a 1 , . . . , a n } is c on taine d in the op en h a lf- sp a c e b e l o w the hyp erplane x d +2 = 0 , then the image by π d +2 of the c onvex hul l of the a i ’s is the c o nvex hul l o f the images o f these p oints, that is, π d +2 (con v( { a 1 , . . . , a n } )) = con v ( { π d +2 ( a 1 ) , . . . , π d +2 ( a n ) } ) . (3) Given any set of p oints, { a 1 , . . . , a n } ⊆ E d +1 , if { a 1 , . . . , a n } is c ontaine d in the op en half-sp ac e ab ove the hyp erpla ne H d +1 or { a 1 , . . . , a n } is c ontaine d in the op en half-sp ac e b elo w H d +1 , then Θ(con v( { a 1 , . . . , a n } )) = con v ( { Θ( a 1 ) , . . . , Θ( a n ) } ) . (4) F or any se t S ⊆ E d +1 , if conv( S ) do es not in terse ct H d +1 , then Θ(con v( S )) = con v (Θ( S )) . Pr o of . (1) The image, π d +2 ( H ), of a h yperplane, H , through the origin is the in tersection of H with H d +2 , whic h is a h yp erplane in H d +2 . (2) This see ms fairly clear g eometric ally but the r esult fails for a r bitra ry sets of p oin ts so to b e on the safe side w e giv e an algebraic pro of. W e will pro v e the followin g t w o facts b y induction on n ≥ 1: 166 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS (1) F or all λ 1 , . . . , λ n ∈ R with λ 1 + · · · + λ n = 1 and λ i ≥ 0, for all a 1 , . . . , a n ∈ H d +2 + (resp. ∈ H d +2 − ) there exis t some µ 1 , . . . , µ n ∈ R with µ 1 + · · · + µ n = 1 and µ i ≥ 0, so that π d +2 ( λ 1 a 1 + · · · + λ n a n ) = µ 1 π d +2 ( a 1 ) + · · · + µ n π d +2 ( a n ) . (2) F or all µ 1 , . . . , µ n ∈ R with µ 1 + · · · + µ n = 1 and µ i ≥ 0, for all a 1 , . . . , a n ∈ H d +2 + (resp. ∈ H d +2 − ) there exis t some λ 1 , . . . , λ n ∈ R with λ 1 + · · · + λ n = 1 and λ i ≥ 0, so that π d +2 ( λ 1 a 1 + · · · + λ n a n ) = µ 1 π d +2 ( a 1 ) + · · · + µ n π d +2 ( a n ) . (1) The base case is clear. Let us assume for the momen t that we prov ed (1 ) for n = 2 and consider the induction step for n ≥ 2. Since λ 1 + · · · + λ n +1 = 1 and n ≥ 2, there is some i suc h that λ i 6 = 1, and without loss of generality , sa y λ 1 6 = 1. Then, w e can write λ 1 a 1 + · · · + λ n +1 a n +1 = λ 1 a 1 + (1 − λ 1 )  λ 2 1 − λ 1 a 2 + · · · + λ n +1 1 − λ 1 a n +1  and since λ 1 + λ 2 + · · · + λ n +1 = 1, w e hav e λ 2 1 − λ 1 + · · · + λ n +1 1 − λ 1 = 1 . By the induction h yp othesis, for n = 2, t here exist α 1 with 0 ≤ α 1 ≤ 1 , suc h t hat π d +2 ( λ 1 a 1 + · · · + λ n +1 a n +1 ) = π d +2  λ 1 a 1 + (1 − λ 1 )  λ 2 1 − λ 1 a 2 + · · · + λ n +1 1 − λ 1 a n +1  = (1 − α 1 ) π d +2 ( a 1 ) + α 1 π d +2  λ 2 1 − λ 1 a 2 + · · · + λ n +1 1 − λ 1 a n +1  Again, by the induction hypothesis (for n ), there exist β 2 , . . . , β n +1 with β 2 + · · · + β n +1 = 1 and β i ≥ 0, so that π d +2  λ 2 1 − λ 1 a 2 + · · · + λ n +1 1 − λ 1 a n +1  = β 2 π d +2 ( a 2 ) + · · · + β n +1 π d +2 ( a n +1 ) , so w e get π d +2 ( λ 1 a 1 + · · · + λ n +1 a n +1 ) = (1 − α 1 ) π d +2 ( a 1 ) + α 1 ( β 2 π d +2 ( a 2 ) + · · · + β n +1 π d +2 ( a n +1 )) = (1 − α 1 ) π d +2 ( a 1 ) + α 1 β 2 π d +2 ( a 2 ) + · · · + α 1 β n +1 π d +2 ( a n +1 ) and clearly , 1 − α 1 + α 1 β 2 + · · · + α 1 β n +1 = 1 as β 2 + · · · + β n +1 = 1; 1 − α 1 ≥ 0 ; and α 1 β i ≥ 0, as 0 ≤ α 1 ≤ 1 a nd β i ≥ 0. This establishes the induction step and th us, all is left is to prov e the case n = 2. 8.5. STERE OGR APHIC PROJECT ION AND THE SP A CE OF SPHERES 167 (2) The base case n = 1 is also clear. As in (1), let us assum e for a momen t that (2) is pro v ed for n = 2 and consider the induction ste p. The pro of is quite simil ar to that of (1) but this time, w e ma y assume that µ 1 6 = 1 and w e write µ 1 π d +2 ( a 1 ) + · · · + µ n +1 π d +2 ( a n +1 ) = µ 1 π d +2 ( a 1 ) + (1 − µ 1 )  µ 2 1 − µ 1 π d +2 ( a 2 ) · · · + µ n +1 1 − µ 1 π d +2 ( a n +1 )  . By the induction hypothesis, there are some α 2 , . . . , α n +1 with α 2 + · · · + α n +1 = 1 and α i ≥ 0 suc h that π d +2 ( α 2 a 2 + · · · + α n +1 a n +1 ) = µ 2 1 − µ 1 π d +2 ( a 2 ) + · · · + µ n +1 1 − µ 1 π d +2 ( a n +1 ) . By the induction h yp othesis for n = 2, there is some β 1 with 0 ≤ β 1 ≤ 1, so that π d +2 ((1 − β 1 ) a 1 + β 1 ( α 2 a 2 + · · · + α n +1 a n +1 )) = µ 1 π d +2 ( a 1 ) + (1 − µ 1 ) π d +2 ( α 2 a 2 + · · · + α n +1 a n +1 ) , whic h establishes the induction h ypot hesis. Therefore, a ll that remains is to pro v e (1) and (2) for n = 2. As π d +2 is giv en b y π d +2 ( x 1 , . . . , x d +2 ) =  x 1 x d +2 , . . . , x d +1 x d +2  ( x d +2 6 = 0) it is enough to treat the case when d = 0, t hat is, π 2 ( a, b ) = a b . T o prov e (1) it is enough to sho w that for an y λ , with 0 ≤ λ ≤ 1, if b 1 b 2 > 0 then a 1 b 1 ≤ (1 − λ ) a 1 + λa 2 (1 − λ ) b 1 + λb 2 ≤ a 2 b 2 if a 1 b 1 ≤ a 2 b 2 and a 2 b 2 ≤ (1 − λ ) a 1 + λa 2 (1 − λ ) b 1 + λb 2 ≤ a 1 b 1 if a 2 b 2 ≤ a 1 b 1 , where, of course (1 − λ ) b 1 + λb 2 6 = 0. F or this, w e compute (lea ving some steps as a n exerci se) (1 − λ ) a 1 + λa 2 (1 − λ ) b 1 + λb 2 − a 1 b 1 = λ ( a 2 b 1 − a 1 b 2 ) ((1 − λ ) b 1 + λb 2 ) b 1 and (1 − λ ) a 1 + λa 2 (1 − λ ) b 1 + λb 2 − a 2 b 2 = − (1 − λ )( a 2 b 1 − a 1 b 2 ) ((1 − λ ) b 1 + λb 2 ) b 2 . 168 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS No w, a s b 1 b 2 > 0, that is, b 1 and b 2 ha v e the same sign and as 0 ≤ λ ≤ 1, w e ha v e b oth ((1 − λ ) b 1 + λb 2 ) b 1 > 0 and ((1 − λ ) b 1 + λb 2 ) b 2 > 0. Then, if a 2 b 1 − a 1 b 2 ≥ 0, that is a 1 b 1 ≤ a 2 b 2 (since b 1 b 2 > 0), the ﬁrst t w o inequalities holds and if a 2 b 1 − a 1 b 2 ≤ 0, that is a 2 b 2 ≤ a 1 b 1 (since b 1 b 2 > 0), the last tw o ineq ualities holds. This pro v es (1 ). In order to pro v e (2), g iven an y µ , with 0 ≤ µ ≤ 1 , if b 1 b 2 > 0, w e sho w that w e can ﬁnd λ with 0 ≤ λ ≤ 1 , so that (1 − µ ) a 1 b 1 + µ a 2 b 2 = (1 − λ ) a 1 + λa 2 (1 − λ ) b 1 + λb 2 . If w e let α = (1 − µ ) a 1 b 1 + µ a 2 b 2 , w e ﬁnd that λ is giv en by the equation λ ( a 2 − a 1 + α ( b 1 − b 2 )) = αb 1 − a 1 . After some (tedious) computations (c hec k fo r y ourself !) we ﬁnd: a 2 − a 1 + α ( b 1 − b 2 ) = ((1 − µ ) b 2 + µb 1 )( a 2 b 1 − a 1 b 2 ) b 1 b 2 αb 1 − a 1 = µb 1 ( a 2 b 1 − a 1 b 2 ) b 1 b 2 . If a 2 b 1 − a 1 b 2 = 0, then a 1 b 1 = a 2 b 2 and λ = 0 w orks. If a 2 b 1 − a 1 b 2 6 = 0, then λ = µb 1 (1 − µ ) b 2 + µb 1 = µ (1 − µ ) b 2 b 1 + µ . Since b 1 b 2 > 0, w e hav e b 2 b 1 > 0, and since 0 ≤ µ ≤ 1, w e conclude that 0 ≤ λ ≤ 1, whic h pro v es (2). (3) Since Θ = π d +2 ◦ b θ ◦ i d +2 , as i d +2 and b θ are linear, they preserv e conv ex h ulls, so b y (2), we simply hav e to sho w that either b θ ◦ i d +2 ( { a 1 , . . . , a n } ) is strictly below the h yperplane, x d +2 = 0, or strictly ab o v e it. But, b θ ( x 1 , . . . , x d +2 ) d +2 = x d +2 − x d +1 and i d +2 ( x 1 , . . . , x d +1 ) = ( x 1 , . . . , x d +1 , 1), so ( b θ ◦ i d +2 )( x 1 , . . . , x d +1 ) d +2 = 1 − x d +1 , and this quantit y is po sitiv e iﬀ x d +1 < 1, negativ e iﬀ x d +1 > 1; that is, either a ll the p oin ts a i are strictly b elo w the h yp erplane H d +1 or all strictly ab o v e it. (4) This follow s immediately fro m (4) as con v( S ) consists of all ﬁnite con v ex combin ations of p oin ts in S .  If a set, { a 1 , . . . , a n } ⊆ E d +2 , contains points on b oth sides of the h yp erplane, x d +2 = 0, then π d +2 (con v ( { a 1 , . . . , a n } )) is not nece ssarily con v ex (ﬁnd suc h a n examp le!). 8.6. STERE OGR APHIC PROJECT ION AND D ELA UNA Y POL YTOPES 169 8.6 Stereograph ic Pro jection, Delaun a y P olytop es and V orono i P olyhedra W e saw in an earlier section that lifting a set of p oin ts, P ⊆ E d , to the parab oloid, P , via the lifting function, l , was fruitful to b etter understand V oronoi diagrams a nd Delauna y triangulations. As far as w e kno w, Edelsbrunner and Seidel [16] w ere the ﬁrst to ﬁnd the relationship betw een V oronoi diagrams and the p olar dual of the con v ex h ull of a lifted set of po ints on to a parab oloid. This connection is described in Note 3.1 of Section 3 in [16]. The connection b etw een the Delauna y triangulation and the con v ex h ull of the lifted set of p oin ts is describ ed in Note 3.2 of the same pap er. P olar dualit y is not men tioned and seems to en ter the scene only with Boissonnat and Yvinec [8]. It t urns out that instead of using a parab oloid w e can use a sphere and instead of the lifting function l w e can use the comp osition of ψ d +1 with the in v erse stereographic pro jection, e τ N . Then, to get bac k do wn to E d , we use the comp o sition of the pro jection, e π N , with π d +1 , instead of the orthogonal pro jection, p d +1 . Ho w ev er, w e ha v e to b e a bit careful b ecaus e Θ do es map a ll conv ex p olyhedra to con v ex p olyhedra. Indeed, Θ is the composition of π d +2 with some linear maps, but π d +2 do es not b eha v e w ell with resp ect to arbitrary con v ex sets . In particular, Θ is no t w ell-deﬁned on an y face that inters ects the h yp erplane H d +1 (of equation x d +1 = 1). F ortunately , w e can circum v en t these diﬃculties b y using the concept of a pro jectiv e p olyhedron in tro duced in Chapter 5. As w e said in the previous section, the corresp ondenc e b et wee n V oronoi diagrams and con v ex hulls via in v ersion was ﬁrst observ ed b y Bro wn [11]. Brow n tak es a set of p o in t s, S , for simplicit y assumed to b e in the plane, ﬁrst lifts these p oin ts to the unit sphere S 2 using in v erse stereographic pro jection (whic h is equiv alen t to an in v ersion of p o w er 2 cen tered at the north p ole), getting τ N ( S ), a nd t hen takes the con v ex h ull, D ( S ) = conv( τ N ( S )), of the lifted set. No w, in order to obtain the V oronoi diagram of S , apply our inv ersion (of p o w er 2 cen tered at the north p ole) to eac h of the faces o f con v( τ N ( S )), obtaining spheres passing through the cen ter o f S 2 and then in tersect these spheres with the plane containing S , o btaining circles. The cen ters of some of these circles are the V oronoi v ertices. Finally , a simple criterion can b e used to retain the “nearest V oro no i p oin ts” and to connect up these v ertices. Note that Bro wn’s metho d is not the method that uses the p olar dual of the p olyhedron D ( S ) = conv( τ N ( S )), as w e migh t ha v e exp ected fro m the lifting metho d using a parab oloid. In fact, it is more natural to get the D elaunay triangulation of S from Bro wn’s metho d, by applying the stereographic pro jection (from the north p ole) to D ( S ), a s w e will prov e b elo w. As D ( S ) is strictly b elo w the plane z = 1, there are no problems. No w, in order to g et the V oronoi diagram, w e ta k e the p olar dual, D ( S ) ∗ , of D ( S ) and then apply the cen tral pro jection w.r.t. the north po le. This is where problems arise, as some faces of D ( S ) ∗ ma y in tersect the h yp erplane H d +1 and this is wh y w e hav e recourse to pro jectiv e geometry . 170 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS First, w e sho w that θ has a go o d b eha vior with respect to tangen t spaces. Recall from Section 5.2 that for an y p oint, a = ( a 1 : · · · : a d +2 ) ∈ P d +1 , the tangent h yp erplane, T a S d , to the sphere S d at a is giv en b y the equation d +1 X i =1 a i x i − a d +2 x d +2 = 0 . Similarly , the ta ngent h yperplane, T a P , to the parab oloid P at a is giv en b y the equation 2 d X i =1 a i x i − a d +2 x d +1 − a d +1 x d +2 = 0 . If we lift a p oin t a ∈ E d to S d b y e τ N ◦ ψ d +1 and to P b y e l , it turns out that the image of the tangen t h yp erplane to S d at e τ N ◦ ψ d +1 ( a ) b y θ is the tangen t hy p erplane to P at e l ( a ). Prop osition 8.7 The map θ has the fol lowing pr op erties: (1) F or any p oint, a = ( a 1 , . . . , a d ) ∈ E d , we have θ ( T e τ N ◦ ψ d +1 ( a ) S d ) = T e l ( a ) P , that i s , θ pr eserves tangent hyp erplanes. (2) F or every ( d − 1) -sp her e, S ⊆ E d , we have θ ( e τ N ◦ ψ d +1 ( S )) = e l ( S ) , that i s , θ pr eserves lifte d ( d − 1) -sphe r es. Pr o of . (1) By Prop osition 8 .5, we kno w that e l = θ ◦ e τ N ◦ ψ d +1 and w e pro v ed in Section 5.2 that pro jectivities preserv e tangen t space s. Th us, θ ( T e τ N ◦ ψ d +1 ( a ) S d ) = T θ ◦ e τ N ◦ ψ d +1 ( a ) θ ( S d ) = T e l ( a ) P , as claimed. (2) This follo ws immediately from the equation e l = θ ◦ e τ N ◦ ψ d +1 . Giv en any tw o distinc t p oin ts, a = ( a 1 , . . . , a d ) and b = ( b 1 , . . . , b d ) in E d , recall that the bisector h yp erplane, H a,b , of a and b is giv en b y ( b 1 − a 1 ) x 1 + · · · + ( b d − a d ) x d = ( b 2 1 + · · · + b 2 d ) / 2 − ( a 2 1 + · · · + a 2 d ) / 2 . W e hav e the follo wing useful prop osition: 8.6. STERE OGR APHIC PROJECT ION AND D ELA UNA Y POL YTOPES 171 Prop osition 8.8 Given any two distinct p oints, a = ( a 1 , . . . , a d ) and b = ( b 1 , . . . , b d ) in E d , the image under the pr oje ction, e π N , of the interse ction , T e τ N ◦ ψ d +1 ( a ) S d ∩ T e τ N ◦ ψ d +1 ( b ) S d , of the tangent hyp erplan e s at the lifte d p oin ts e τ N ◦ ψ d +1 ( a ) and e τ N ◦ ψ d +1 ( b ) on the spher e S d ⊆ P d +1 is the em b e dding of the bise ctor hyp erplan e, H a,b , of a and b , in to P d , that is, e π N ( T e τ N ◦ ψ d +1 ( a ) S d ∩ T e τ N ◦ ψ d +1 ( b ) S d ) = ψ d +1 ( H a,b ) . Pr o of . In view of the geometric interpre tation of e π N giv en earlier, w e need to ﬁnd the equation of the h yp erplane, H , passing through the in tersection of the tangen t h yp erplan es, T e τ N ◦ ψ d +1 ( a ) and T e τ N ◦ ψ d +1 ( b ) and passin g through the north p ole and then, it is geometrically ob vious that e π N ( T e τ N ◦ ψ d +1 ( a ) S d ∩ T e τ N ◦ ψ d +1 ( b ) S d ) = H ∩ H d +1 (0) , where H d +1 (0) is the h yp erplane (in P d +1 ) of equation x d +1 = 0. Recall that T e τ N ◦ ψ d +1 ( a ) S d and T e τ N ◦ ψ d +1 ( b ) S d are giv en by E 1 = 2 d X i =1 a i x i + ( d X i =1 a 2 i − 1) x d +1 − ( d X i =1 a 2 i + 1) x d +2 = 0 and E 2 = 2 d X i =1 b i x i + ( d X i =1 b 2 i − 1) x d +1 − ( d X i =1 b 2 i + 1) x d +2 = 0 . The h yperplanes passing through T e τ N ◦ ψ d +1 ( a ) S d ∩ T e τ N ◦ ψ d +1 ( b ) S d are giv en b y an equation of the form λE 1 + µE 2 = 0 , with λ, µ ∈ R . F urthermore, in order to contain the north p ole, this eq uation m ust v anish for x = (0 : · · · : 0 : 1 : 1). But, observ e that setting λ = − 1 and µ = 1 giv es a solution since the correspo nding equation is 2 d X i =1 ( b i − a i ) x i + ( d X i =1 b 2 i − d X i =1 a 2 i ) x d +1 − ( d X i =1 b 2 i − d X i =1 a 2 i ) x d +2 = 0 and it v anishes on (0 : · · · : 0 : 1 : 1). But then, the interse ction of H with the hyperplane x d +1 = 0 is giv en by 2 d X i =1 ( b i − a i ) x i − ( d X i =1 b 2 i − d X i =1 a 2 i ) x d +2 = 0 , whic h is equiv alen t to the equation of ψ d +1 ( H a,b ) (except that x d +2 is replaced b y x d +1 ). In order to deﬁne precisely Delauna y complex es a s pro jections o f ob jects obtained by deleting some faces f rom a pro jectiv e p olyhedron w e need to deﬁne the notion of “pro jectiv e (p olyhedral) complex”. How ev er, this is easily done b y deﬁning the notion of cell complex where the cells are p olyhedral cones. Suc h ob jects are kno wn a s fa ns . The deﬁnition b elo w is basically Deﬁnition 6.8 in whic h the cells are cones as opp osed to polytop es. 172 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Deﬁnition 8.5 A fan in E m is a set, K , consisting of a (ﬁnite or inﬁnite) set of p olyhedral cones in E m satisfying the follo wing conditions: (1) Eve ry face of a cone in K also b elongs to K . (2) F or any t w o cones σ 1 and σ 2 in K , if σ 1 ∩ σ 2 6 = ∅ , then σ 1 ∩ σ 2 is a common face o f b oth σ 1 and σ 2 . Ev ery cone, σ ∈ K , of dimens ion k , is called a k -fac e (or fac e ) of K . A 0-face { v } is called a vertex and a 1-fa ce is called an e dge . The dim ension o f the fan K is the maxim um of the dimensions of all cones in K . If dim K = d , then ev ery face of dimension d is called a c el l and ev ery face of dimension d − 1 is called a fac et . A pr o je ctive (p olyhe dr al) c o m plex , K ⊆ P d , is a set of pro jectiv e p o lyhe dra of the form, { P ( C ) | C ∈ K } , where K ⊆ R d +1 is a fan. Giv en a pro jectiv e complex, the notions of face, v ertex, edge, cell, facet, are dedine d in the ob vious w ay . If K ⊆ R d is a p olyhe dral complex, then it is easy to che ck that the set { C ( σ ) | σ ∈ K } ⊆ R d +1 is a fan and w e get the pro jectiv e complex e K = { P ( C ( σ )) | σ ∈ K } ⊆ P d . The pro jectiv e complex, e K , is called the pr oje ctive c ompletion of K . Also, it is easy to c hec k that if f : P → P ′ is an injectiv e aﬃne map b etw een tw o p olyhedra P and P ′ , then f extends uniquely to a pro jectivit y , e f : e P → f P ′ , b et w een the pro jectiv e completions of P and P ′ . W e now ha v e all the facts needed to sho w that Delauna y triangulations and V oronoi diagrams can b e deﬁne d in terms of the lifting, e τ N ◦ ψ d +1 , and the pro jection, e π N , and to establish their dualit y via p olar dualit y with resp ect to S d . Deﬁnition 8.6 Giv en any set of p oin ts, P = { p 1 , . . . , p n } ⊆ E d , the polytop e, D ( P ) ⊆ R d +1 , called the De launay p o l ytop e asso ciated with P is the con v ex h ull of the union of the lifting of the p oin ts of P on to the sphere S d ( via in v erse stereographic pro jection) with the north p ole, that is, D ( P ) = conv( τ N ( P ) ∪ { N } ). The pr oje ctive Delaunay p ol ytop e , e D ( P ) ⊆ P d +1 , asso ciated with P is the pro jectiv e completion of D ( P ). The p olyhedral complex, C ( P ) ⊆ R d +1 , called the lifte d Delaunay c ompl e x of P is t he complex obtained from D ( P ) b y deleting the facets con taining the north p ole (and their faces) and e C ( P ) ⊆ P d +1 is the pro jectiv e completion of C ( P ). The p olyhedral complex, D el ( P ) = π d +1 ◦ e π N ( e C ( P )) ⊆ E d , is the Delaunay c omplex of P o r Delaunay triangulation of P . The ab o v e is not the “standard” deﬁnition of the D elauna y triangulation of P but it is equiv alen t to the deﬁnition, say g iv en in Boissonnat and Yvinec [8], as w e will prov e shortly . It also has certain adv an tages o v er lifting on to a parab oloid, as w e will explain. 8.6. STERE OGR APHIC PROJECT ION AND D ELA UNA Y POL YTOPES 173 It it p ossible and useful to deﬁne D el ( P ) more directly in terms of C ( P ). The pro jection, f π N : ( P d +1 − { N } ) → P d , comes from the linear map, b π N : R d +2 → R d +1 , giv en by b π N ( x 1 , . . . , x d +1 , x d +2 ) = ( x 1 , . . . , x d , x d +2 − x d +1 ) . Conseque ntly , as e C ( P ) = ] C ( P ) = P ( C ( C ( P ))), w e immediately c hec k that D el ( P ) = π d +1 ◦ e π N ( e C ( P )) = π d +1 ◦ b π N ( C ( C ( P )) ) = π d +1 ◦ b π N (cone( [ C ( P ))) , where [ C ( P ) = { b u | u ∈ C ( P ) } and b u = ( u, 1). This suggests deﬁning the map, π N : ( R d +1 − H d +1 ) → R d , b y π N = π d +1 ◦ b π N ◦ i d +2 , whic h is explicit y give n b y π N ( x 1 , . . . , x d , x d +1 ) = 1 1 − x d +1 ( x 1 , . . . , x d ) . Then, as C ( P ) is strictly b elo w the h yp erplane H d +1 , w e ha v e D el ( P ) = π d +1 ◦ e π N ( e C ( P )) = π N ( C ( P )) . First, note that D el ( P ) = π d +1 ◦ e π N ( e C ( P )) is indee d a p olyhedral complex whose geo- metric realization is the con v ex h ull, con v ( P ), of P . Indeed, by Prop osition 8 .6 , the images of the facets o f C ( P ) are p olytop es and when an y tw o suc h p olytop es meet, they meet along a common face. F urthermore, if dim(con v( P )) = m , then D el ( P ) is pure m -dimens ional. First, D el ( P ) con tains at least one m - dimensional cell. If D el ( P ) w as not pure, as the complex is connected there w ould b e some cell, σ , of dimension s < m meeting some o ther cell, τ , of dime nsion m along a common face of dimension at most s and b ecause σ is not con tained in any face of dimension m , no facet of τ con taining σ ∩ τ can b e adjacen t to a n y cell of dimension m and so, D el ( P ) w ould not be con v ex, a con tradiction. F o r an y p olytop e, P ⊆ E d , giv en an y p oin t, x , not in P , recall that a facet, F , of P is visible fr om x iﬀ for ev ery po in t , y ∈ F , the line through x and y interse cts F only in y . If dim( P ) = d , this is equiv alen t to sa ying that x and the inte rior o f P are strictly separated b y the supp orting h yp erplane of F . Note that if dim ( P ) < d , it p ossible that ev ery facet of P is visible from x . No w, assume that P ⊆ E d is a p olytop e with nonemp ty in terior. W e sa y that a facet, F , of P is a lower-facing fa c et of P iﬀ the unit normal to the supp orting h yp erplane of F p oin ting tow a rds the in terior of P has non-negativ e x d +1 -co ordinate. A facet, F , that is not lo w er-facing is called an upp er-fa c i ng fac et (Note that in this case the x d +1 co ordinate of the unit normal to the supp orting h yp erplane of F p oin ting to w ards the in terior o f P is strictly negativ e). Here is a con v enien t w a y to c haracterize lo w er-facing facets. 174 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Prop osition 8.9 Given any p olytop e, P ⊆ E d , with nonem pty interior, for any p oint, c , on the O x d -axis, if c lies strictly ab ove al l the interse ction p oints of the O x d -axis with the supp orting hyp erplanes of al l the upp er-faci n g fac ets of F , then the lower-facing fac ets of P ar e exactly the fac ets not visible fr om c . Pr o of . Note that the in tersection p oin ts of the O x d -axis with the supporting h yp erplanes of all the upp er-facing facets of P are strictly ab o v e the in tersection p oin ts of the O x d -axis with the supporting h yp erplanes of all the lo w er-facing facets. Supp ose F is visible from c . Then, F m ust not b e low er-fa cing as otherwise, for any y ∈ F , the line through c and y has to in tersect some upp er-facing facet and F is not b e visible from c , a con tradiction. No w, as P is the in tersection of the closed half- space s determined b y the supp orting h yp erplanes of its facets, b y the deﬁnition of an upp er-facing facet, an y p oin t, c , on the O x d - axis that lies strictly ab o ve the the in tersection p oin ts of the O x d -axis with the supporting h yp erplanes of all the upp er- facing facets of F has the prop ert y that c and the interior of P are strictly separated b y all these supp orting h yp erplanes. T herefore, all the upp er- facing facets of P are visible from c . It follo ws that the facets visib le from c are exactly the upp er-facing facets, as claimed. W e will also need the following fact when dim( P ) < d . Prop osition 8.10 Given any p o lytop e, P ⊆ E d , ther e is a p oint, c , on the O x d -axis, s uch that for al l p oints, x , on the O x d -axis and ab ove c , the set o f fa c ets of con v( P ∪ { x } ) not c on tain ing x is identic al. Mor e ove r, the set of fac ets o f P not v isible fr om x is the set of fac ets of con v ( P ∪ { x } ) that do not c o n tain x . Pr o of . If dim( P ) = d then pick a n y c on the O x d -axis ab o v e the in tersection p oin ts of the O x d -axis with the suppo rting h yp erplanes of a ll the upper-fa cin g facets o f F . Then, c is in general p osition w.r.t. P in the sense that c and any d v ertices of P do not lie in a common h yp erplane. No w, our result follo ws by lemma 8.3.1 of Boissonnat and Yvinec [8]. If dim( P ) < d , consider the aﬃne hu ll o f P with the O x d +1 -axis and use the same argumen t. Corollary 8.11 Given any p ol ytop e, P ⊆ E d , with nonempty interior, ther e is a p oint, c , on the O x d -axis, so that for al l x o n the O x d -axis and a b ove c , the lower-facing fac ets of P ar e exactly the fac ets of conv( P ∪ { x } ) that do not c ontain x . As usual, let e d +1 = (0 , . . . , 0 , 1) ∈ R d +1 . Theorem 8.12 Given any set of p o i nts, P = { p 1 , . . . , p n } ⊆ E d , let D ′ ( P ) denote the p ol yhe dr on con v ( l ( P )) + cone( e d +1 ) and let e D ′ ( P ) b e the pr oje ctive c ompletion of D ′ ( P ) . A lso, let C ′ ( P ) b e the p olyhe dr al c omplex c onsisting of the b ounde d fac ets of the p olytop e D ′ ( P ) and let e C ′ ( P ) b e the p r oje ctive c ompletion of C ′ ( P ) . The n θ ( e D ( P )) = e D ′ ( P ) a n d θ ( e C ( P )) = e C ′ ( P ) . 8.6. STERE OGR APHIC PROJECT ION AND D ELA UNA Y POL YTOPES 175 F urthermor e, if D el ′ ( P ) = π d +1 ◦ e p d +1 ( e C ′ ( P )) = p d +1 ( C ′ ( P )) is the “standar d” Delaunay c om plex of P , that is, the ortho g o nal pr oje ction of C ′ ( P ) onto E d , then D el ( P ) = D el ′ ( P ) . Ther efor e, the two notions o f a Delaunay c omplex agr e e. I f dim(con v ( P )) = d , then the b ounde d fac ets of conv( l ( P )) + cone( e d +1 ) ar e p r e c isely the lo wer-facing fac ets of con v ( l ( P )) . Pr o of . Recall that D ( P ) = con v( τ N ( P ) ∪ { N } ) and e D ( P ) = P ( C ( D ( P ))) is the pro jectiv e completion of D ( P ). If w e write \ τ N ( P ) for { \ τ N ( p i ) | p i ∈ P } , then C ( D ( P )) = cone( \ τ N ( P ) ∪ { b N } ) . By deﬁnition, w e ha v e θ ( e D ) = P ( b θ ( C ( D ))) . No w, as b θ is linear, b θ ( C ( D )) = b θ (cone( \ τ N ( P ) ∪ { b N } )) = cone( b θ ( \ τ N ( P )) ∪ { b θ ( b N ) } ) . W e claim that cone( b θ ( \ τ N ( P )) ∪ { b θ ( b N ) } ) = cone( d l ( P ) ∪ { (0 , . . . , 0 , 1 , 1 ) } ) = C ( D ′ ( P )) , where D ′ ( P ) = con v( l ( P )) + cone( e d +1 ) . Indeed, b θ ( x 1 , . . . , x d +2 ) = ( x 1 , . . . , x d , x d +1 + x d +2 , x d +2 − x d +1 ) , and for an y p i = ( x 1 , . . . , x d ) ∈ P , \ τ N ( p i ) = 2 x 1 P d i =1 x 2 i + 1 , . . . , 2 x d P d i =1 x 2 i + 1 , P d i =1 x 2 i − 1 P d i =1 x 2 i + 1 , 1 ! = 1 P d i =1 x 2 i + 1 2 x 1 , . . . , 2 x d , d X i =1 x 2 i − 1 , d X i =1 x 2 i + 1 ! , so w e get b θ ( \ τ N ( p i )) = 2 P d i =1 x 2 i + 1 x 1 , . . . , x d , d X i =1 x 2 i , 1 ! = 2 P d i =1 x 2 i + 1 d l ( p i ) . 176 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS Also, w e hav e b θ ( b N ) = b θ (0 , . . . , 0 , 1 , 1) = (0 , . . . , 0 , 2 , 0) = 2 d e d +1 , and b y deﬁnition of cone( − ) (scalar factors are irrelev an t), w e get cone( b θ ( \ τ N ( P )) ∪ { b θ ( b N ) } ) = cone( d l ( P ) ∪ { (0 , . . . , 0 , 1 , 1) } ) = C ( D ′ ( P )) , with D ′ ( P ) = con v( l ( P )) + cone( e d +1 ), as claimed. This prov es that θ ( e D ( P )) = e D ′ ( P ) . No w, it is clear that the facets of con v( τ N ( P ) ∪ { N } ) that do not contain N are mapped to the b ounded facets of con v( l ( P )) + cone( e d +1 ), since N go es the p oin t at inﬁnit y , so θ ( e C ( P )) = e C ′ ( P ) . As e π N = e p d +1 ◦ θ b y Prop osition 8.5, w e get D el ′ ( P ) = π d +1 ◦ e p d +1 ( e C ′ ( P )) = π d +1 ◦ ( e p d +1 ◦ θ )( e C ( P )) = π d +1 ◦ e π N ( e C ( P )) = D el ( P ) , as claimed. Finally , if dim(con v ( P )) = d , then, b y Corollary 8.11, we can pic k a p oin t, c , on the O x d +1 -axis, so that the facets o f con v( l ( P ) ∪ { c } ) that do not con tain c are precisely the lo w er-facing facets of conv( l ( P )). How ev er, it is also clear that the f a cets of con v( l ( P ) ∪ { c } ) that contain c tend to the un b ounded facets of D ′ ( P ) = conv( l ( P )) + cone( e d +1 ) when c go es to + ∞ . W e can also c haracterize when the D elauna y complex, D el ( P ), is simplicial. Recall that w e sa y that a set of po ints, P ⊆ E d , is in gener al p osition iﬀ no d + 2 of the p oin ts in P b elong to a common ( d − 1)-sphere. Prop osition 8.13 Given any set of p o i nts, P = { p 1 , . . . , p n } ⊆ E d , if P is i n gener al p os i tion , then the Delaunay c omplex, D el ( P ) , is a pur e si m plicial c omplex. Pr o of . Let dim(con v ( P )) = r . Then, τ N ( P ) is con tained in a ( r − 1)- sphe re of S d , so w e ma y assume tha t r = d . Suppose D el ( P ) has some facet, F , whic h is not a d -simplex. If so, F is the con v ex h ull of at least d + 2 p oin ts, p 1 , . . . , p k of P and since F = π N ( b F ), for some facet, b F , of C ( P ), w e deduce tha t τ N ( p 1 ) , . . . , τ N ( p k ) b elong to the supp orting hy p erplane, H , o f b F . Now, if H passes through the north p ole, then w e kno w t hat p 1 , . . . , p k b elong to some h yp erplane of E d , whic h is imp ossible since p 1 , . . . , p k are the ve rtices of a facet of dimension d . Th us, H do es not pass t hro ugh N and so, p 1 , . . . , p k b elong to some ( d − 1)-sphere in E d . As k ≥ d + 2, this con tradicts the assu mption that the po in t s in P are in general p osition. Remark: Ev en when the p oints in P are in g ene ral p osition, the D elauna y p olytop e, D ( P ), ma y not b e a simplicial p olytope. F or example , if d + 1 p oin ts b elong to a h yp erplane in 8.6. STERE OGR APHIC PROJECT ION AND D ELA UNA Y POL YTOPES 177 E d , then the lifted p oin ts b elong to a hyperplane passing through the north p ole and these d + 1 lifted p oin ts together with N may form a non-simplicial facet. F or example, consider the p olytop e obtained b y lifting our original d + 1 p o in ts on a h yp erplane , H , plus one more p oin t not in the the h yp erplane H . W e can also c haracterize the V oronoi diagra m of P in terms o f the p olar dual of D ( P ). Unfortunately , w e can’t simply tak e the p olar dual, D ( P ) ∗ , of D ( P ) and pro ject it using π N b ecause some of the facets of D ( P ) ∗ ma y in tersect the h yp erplane, H d +1 , and π N is undeﬁ ned on H d +1 . Ho w ev er, using pro jectiv e completions, w e can indeed reco v er the V oronoi diagram of P . Deﬁnition 8.7 Giv en an y set of p oints , P = { p 1 , . . . , p n } ⊆ E d , the V or onoi p olyhe dr on asso ciated with P is the p olar dual (w.r.t. S d ⊆ R d +1 ), V ( P ) = ( D ( P )) ∗ ⊆ R d +1 , of the Delaunay p olytop e, D ( P ) = con v ( τ N ( P ) ∪ { N } ). The pr oj e ctive V or onoi p olytop e , e V ( P ) ⊆ P d +1 , asso ciated with P is the pro jectiv e completion of V ( P ). The p olyhedral complex, V or ( P ) = π d +1 ◦ e π N ( e V ( P )) ⊆ E d , is the V or onoi c omplex of P or V or onoi diagr am of P . Giv en an y set of p oints , P = { p 1 , . . . , p n } ⊆ E d , let V ′ ( P ) = ( D ′ ( P )) ∗ b e the pola r dual (w.r.t. P ⊆ R d +1 ) of the “standard” Delauna y polyhedron deﬁn ed in Theorem 8.12 and let e V ′ ( P ) = ^ V ′ ( P ) ⊆ P d b e its pro jectiv e completion. It is not hard to che c k that p d +1 ( V ′ ( P )) = π d +1 ◦ e p d +1 ( e V ′ ( P )) is the “standard” V oro noi diagram, denoted V o r ′ ( P ). Theorem 8.14 Given any set of p oin ts, P = { p 1 , . . . , p n } ⊆ E d , we have θ ( e V ( P )) = e V ′ ( P ) and V or ( P ) = V or ′ ( P ) . Ther efor e, the two notions of V or onoi di a gr ams agr e e. Pr o of . By deﬁnition, e V ( P ) = ] V ( P ) = ^ ( D ( P )) ∗ and b y Prop osition 5.12, ^ ( D ( P )) ∗ =  ] D ( P )  ∗ = ( e D ( P )) ∗ , so e V ( P ) = ( e D ( P )) ∗ . By Prop osition 5.10, θ ( e V ( P )) = θ (( e D ( P )) ∗ ) = ( θ ( e D ( P ))) ∗ 178 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS and b y Theorem 8.12, θ ( e D ( P )) = e D ′ ( P ) , so w e get θ ( e V ( P )) = ( e D ′ ( P )) ∗ . But, b y Prop osition 5.12 again, ( e D ′ ( P )) ∗ =  ^ D ′ ( P )  ∗ = ^ ( D ′ ( P )) ∗ = ^ V ′ ( P ) = e V ′ ( P ) . Therefore, θ ( e V ( P )) = e V ′ ( P ) , as claimed. As e π N = e p d +1 ◦ θ b y Prop osition 8.5, w e get V or ′ ( P ) = π d +1 ◦ e p d +1 ( e V ′ ( P )) = π d +1 ◦ e p d +1 ◦ θ ( e V ( P )) = π d +1 ◦ e π N ( e V ( P )) = V or ( P ) , ﬁnishing the pro of. W e can also prov e the prop osition b elo w whic h sho ws directly that V or ( P ) is the V oronoi diagram of P . R ecall that that e V ( P ) is the pro j ectiv e completion of V ( P ). W e o bse rv ed in Section 5.2 (see pag e 78) that in the patch U d +1 , there is a bijection b etw een the faces of e V ( P ) and t he faces of V ( P ). F urthermore, the pro jectiv e completion, e H , of ev ery h yperplane, H ⊆ R d , is also a h yperplane a nd it is easy to see that if H is tangen t to V ( P ), then e H is tangen t to e V ( P ). Prop osition 8.15 Given any set of p oints, P = { p 1 , . . . , p n } ⊆ E d , for every p ∈ P , i f F is the fac et of V ( P ) that c ontains τ N ( p ) , if H is the tangent hyp erplane at τ N ( p ) to S d and if F is cut out by the hyp erplane s H , H 1 , . . . , H k p , in the sense that F = ( H ∩ H 1 ) − ∩ · · · ∩ ( H ∩ H k p ) − , wher e ( H ∩ H i ) − denotes the close d hal f - sp ac e in H c ontaini n g τ N ( p ) determine d by H ∩ H i , then V ( p ) = π d +1 ◦ e π N ( e H ∩ e H 1 ) − ∩ · · · ∩ e π N ( e H ∩ e H k p ) − is the V or on o i r e gion of p (wher e π d +1 ◦ e π N ( e H ∩ e H i ) − is the close d half-sp ac e c o n taining p ). If P is in g ener al p o s ition, then V ( P ) is a simple p olyhe dr o n (every vertex b elon gs to d + 1 fac ets). 8.7. APPLIC A TIONS 179 Pr o of . Recall that b y Prop osition 8.5, e τ N ◦ ψ d +1 = ψ d +2 ◦ τ N . Eac h H i = T τ N ( p i ) S d is the tangen t h yp erplane to S d at τ N ( p i ), for some p i ∈ P . Now, b y deﬁnition of the pro jectiv e completion, the em b edding, V ( P ) − → e V ( P ), is giv en b y a 7→ ψ d +2 ( a ). Thus , ev ery p oin t, p ∈ P , is mapp ed to the p oin t ψ d +2 ( τ N ( p )) = f τ N ( ψ d +1 ( p )) and w e also ha ve e H i = T e τ N ◦ ψ d +1 ( p i ) S d and e H = T e τ N ◦ ψ d +1 ( p ) S d . By Prop osition 8.8, e π N ( T e τ N ◦ ψ d +1 ( p ) S d ∩ T e τ N ◦ ψ d +1 ( p i ) S d ) = ψ d +1 ( H p,p i ) is the em b edding of the bisector h yp erplane of p and p i in P d , so the ﬁrst part holds. No w, assume that some v ertex, v ∈ V ( P ) = D ( P ) ∗ , b elongs to k ≥ d + 2 facets of V ( P ). By p olar dualit y , this means t hat the facet, F , dual of v has k ≥ d + 2 v ertices τ N ( p 1 ) , . . . , τ N ( p k ) of D ( P ). W e claim that τ N ( p 1 ) , . . . , τ N ( p k ) m ust b elong t o some hy - p erplane passing through the north p ole. Otherwis e, τ N ( p 1 ) , . . . , τ N ( p k ) would belong to a h yp erplane not passing through the north p ole and so they wo uld b elong to a ( d − 1) sphere of S d and th us, p 1 , . . . , p k w ould b elong to a ( d − 1)-sphere ev en though k ≥ d + 2, con tradicting that P is in general p osition. But then, by p olar dualit y , v w ould b e a point at inﬁnit y , a contradiction. Note that whe n P is in general p osition, ev en though the p olytop e, D ( P ), ma y not be simplicial, its dual, V ( P ) = D ( P ) ∗ , is a simple p olyhe dr on . What is happ ening is that V ( P ) has unbounded faces whic h ha v e “v ertices at inﬁnity ” that do not coun t! In fact, the fa ces of D ( P ) that fail to be simplicial are those that are contained in some h yperplane through the north p ole. By po lar duality , these faces corresp ond to a v ertex at inﬁnit y . Also, if m = dim(con v ( P )) < d , then V ( P ) ma y not ha v e any v ertices! W e conclude our presen tation of V or o noi diag rams and Delauna y triangulations with a short section on applications. 8.7 Applications of V o ronoi Diagrams and D e launa y T riangulations The examples b elo w are take n from O’Rourke [29]. Other examples can b e f ound in Preparata and Shamos [30], Boissonnat a nd Yvinec [8], a nd de Berg, V an Krev eld, Ov ermars, and Sc h w arzk opf [5]. The ﬁrst example is the ne ar est neighb ors problem. There are actually tw o subproblems: Ne ar est neighb or queries and al l ne ar est ne i g hb ors . The nearest neighbor queries problem is as follo ws. Given a set P of p oin ts and a query p oin t q , ﬁnd the nearest neighbor(s) of q in P . This problem can b e solve d b y computing the V oronoi diagram of P and determining in whic h V oronoi region q falls. This last problem, 180 CHAPTER 8. D IRICHLE T–VOR O NOI DIA GRAMS called p oin t lo c a tion , has b een heavily studied (see O’Rourk e [29 ]). The all neigh b ors problem is as fo llo ws: Giv en a set P of p oin ts, ﬁnd the nearest neigh b or(s) to all p oin ts in P . This problem can b e solv ed b y building a graph, the ne ar est neighb or gr aph , for short nng . The no des of this undirected g raph a re the po ints in P , and t here is an arc from p to q iﬀ p is a nearest neigh b or of q or vice v ersa. Then it can b e show n that this graph is con tained in the Delauna y triangulation of P . The second example is the lar g e s t empty cir cle . Some practical applications of t his problem a re to lo cate a new store (to a v oid comp etition), or to lo cate a n uclear plan t as far as po ssible from a set of to wns. More precisely , the problem is as fo llows. Giv en a set P of p oin ts, ﬁnd a largest empty circle whose cente r is in the (closed) conv ex h ull of P , empt y in that it contains no p oin ts fr o m P inside it, and largest in the sense that there is no other circle with strictly larger radius. The V oronoi diagram of P can be used to solv e this problem. It can b e sho wn that if the cente r p o f a largest empt y circle is strictly inside the con v ex hu ll of P , then p coincides with a V oronoi v ertex. How ev er, not ev ery V oro noi v ertex is a go o d candidate. It can also be sho wn that if the cente r p of a largest empt y circle lies on the b oundary of the con v ex h ull of P , then p lies on a V or o noi edge. The third example is the mini m um sp annin g tr e e . Giv en a graph G , a minim um spanning tree of G is a subgraph of G that is a tree, con tains eve ry v ertex of the graph G , and minimizes the sum of the lengths of the tree edges. It can b e shown that a minim um spanning tree is a subgraph of the Delaunay triangulation of the v ertices of the graph. This can b e used to impro v e algorithms for ﬁnding minim um spanning trees, for example Krusk al’s algorithm (see O’Rourk e [29]). W e conclude by mentioni ng that V oronoi diagrams hav e applications to m otion pla n ning . F o r example, consider the problem of movi ng a disk on a plane while a v oiding a set o f p olygonal obstacles. If w e “extend” the obstacles by the diameter of the disk, the problem reduces to ﬁnding a collision–free path b etw een t w o p oin ts in the extended obstacle space. One need s to generalize the notion of a V oronoi diagram. Indeed, w e need to deﬁne the distance to an ob ject, and medial curv es (consisting of p oin ts equ idistan t to tw o ob jects) ma y no longer b e straigh t lines. A collision–free path with maximal clearance from the obstacles can b e fo und b y mo ving a lo ng the edges of the generalized V oronoi diagram. This is an activ e area of researc h in rob otics. F or more on this topic, see O’Rourk e [29]. A cknow le dgement . I wis h to thank Marcelo Sique ira for suggesting man y impro v emen ts and for catc hing man y bugs with his “eagle ey e”. 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Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

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