Zone Diagrams in Euclidean Spaces and in Other Normed Spaces

ZONE DIA GRAMS IN EUCLIDEAN SP A CES AND IN OTHER NORMED SP A CES Akitoshi Ka w amura 1 Department of Computer Science, U niversi ty of T o ronto 10 King’s College Road, T oronto, Ontario , M5S 3G4 Canada kawamura@c s.toronto.ed u Ji ˇ r ´ ı Ma to u ˇ sek Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles Universit y Malostransk ´ e n´ am. 25, 118 00 Praha 1, Czech R ep ublic, and Institute of Theoretical Computer Science, ETH Z ¨ urich 8092 Z ¨ urich, Switzerland matousek@k am.mff.cuni. cz T akeshi Tokuy a ma 2 Graduate S chool of Information Sciences, T ohok u Universit y Aramaki Aza A oba, A oba-ku, Sendai, 980-8579 Japan tokuyama@d ais.is.tohok u.ac.jp Abstract. Zone diagram is a v ariation on the cla ssical concept of a V oronoi diagr am. Given n sites in a metric space that co mpete for territory , the zone diagra m is an equilibr ium state in the comp etition. F orma lly it is defined as a fixed p oint of a ce r tain “do minance” map. Asano, Matou ˇ s ek, and T okuyama prov ed the exis tence and uniquenes s o f a zone diagram for po int sites in Euclidea n plane, and Reem and Reich show ed exis tence for tw o arbitrar y sites in an arbitr ary metric space. W e establis h existence a nd uniqueness for n disjoint compact sites in a Euclidean space of arbitr ary (finite) dimension, and mo re g enerally , in a finite-dimens io nal normed spa ce with a smooth and r otund norm. The pro o f is considerably simpler than tha t of Asano et al. W e also provide a n exa mple of non-uniqueness for a nor m that is rotund but not smo oth. Finally , we pr ove existence and uniqueness for tw o p oint sites in the plane with a smo oth (but not necessa rily rotund) no rm. 1 P art of this work w as done while A.K. w as v isiting ETH Z ¨ uric h, whose supp ort and h ospitalit y are grate- fully ackno wledged. His research is also supp orted by the Nakajima F oundation and t h e Natural Sciences and Engineering Researc h Council of Canada. 2 The part of this research by T.T. wa s p artially supp orted by the JSPS Grant-in-Aid for Scientific Research (B) 183000 01. KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 1 Figure 1. A zone diagram of p oin ts and segmen ts. 1. Introduction Zone diagram is a metric notion somewhat similar to the classical concept of a V oronoi diagram. Let ( X, dist ) b e a metric space and let P = ( P 1 , . . . , P n ) b e an n -tuple of n onempt y subsets of X called the sites . T o a vo id un pleasan t trivialiti es, w e will alw ays assume in this pap er th at th e sites are closed and pairwise disj oin t. A zone diagr am of the n -tuple P is an n -tuple R = ( R 1 , . . . , R n ) of s u bsets of X , called th e r e gions of the zone diagram, with th e follo wing defining prop ert y: Eac h R i consists of all p oin ts x ∈ X that are closer (non-strictly) to P i than to the union S j 6 = i R j of all the other regions. Fig. 1 shows a zone diagram in Euclidean plane whose s ites are p oin ts and segmen ts. While in t he V oronoi diag ram the reg ions partition the wh ole space, in a zone diagram the u nion of the regions t ypically h as a nonempt y complemen t, called the neutr al zone . The definition of the zone diagram is implicit, since eac h r egion is determined in terms of the remaining ones. S o neither existence n or uniqu eness of the zone d iagram is ob vious, an d so far only p artial r esults in this direction ha ve b een kno wn. Asano et al. [2] introdu ced the notion of a zone diagram, for the case of n p oint sites in Euclidean plane, and in this setting they pr o ved existence and u niqueness. Th e pro of inv olv es a case analysis sp ecific to R 2 . Reem a nd Reic h [8] established , by a simp le and ele gan t argument, t he existence of a zone diagram for two sites in an arbitrary metric sp ace (and ev en in a still more general setting, whic h they call m -sp ac es ). On the negativ e side, they ga v e an example of a t hree-p oint metric space in wh ich the z one diagram of t wo p oint site s is not uniqu e; th us, uniqueness needs add itional assumptions. On the other hand, f or all we kno w, it is p ossible that a zone diagram alw a ys exists, f or arbitrary sites in an arbitrary metric space. Arbitrary sites in Euclidean spaces. In this pap er, we establish the existence and u nique- ness of zone diagrams in Euclidean spaces. This generaliz es the main result of [2] with a considerably simpler argumen t. F or the ca se of t w o p oin t sites in th e plane, w e also obtain a new and simpler pro of of the existence and uniqueness of the distanc e trise ctor curve consid ered b y Asano et al. [3]. Theorem 1.1. L et the c onsider e d metric sp ac e ( X , dist) b e R d with the Euclide an distanc e. F or every n -tuple P = ( P 1 , . . . , P n ) of nonempty close d sites in R d such that dist( P i , P j ) > 0 for every i 6 = j , ther e exists exactly one zone diagr am R . The full pro of is con tained in S ections 2 (general p reliminaries) and 3. T he same pro of yields existence and uniqueness also for infinitely man y sites in R d , p ro vided that eve ry t wo of t hem ha v e distance at least 1 (or some fixed ε > 0). Moreo ve r, with some extra effort it ma y b e 2 ZONE DIAGRAMS IN N ORMED SP ACES P 1 = { (0 , 0) } P 2 = { (0 , 3) } R 1 R 2 P 1 = { (0 , 0) } P 2 = { (0 , 3) } R 1 R 2 Figure 2. Two different zone d iagrams un d er the ℓ 1 metric (drawn in the grid with u nit s p acing). p ossible to extend the p ro of to compact sites in a Hilb ert space, f or example, bu t in this pap er w e restrict our selv es to the finite-dimensional setting. Normed spaces. W e also inv estigate zone d iagrams in a more general class of metric spaces, namely , finite-dimensional norm ed spaces. 3 Normed sp aces are among the m ost imp ortant classes of metric spaces. Moreo v er, as w e will see, studying arbitrary norms also s heds some ligh t on the Euclidean case. Earlier Asano and Kirkpatric k [1] inv estigated distance trisector curv es (whic h are essenti ally equiv alen t to t w o-site zone diagrams) of tw o p oint sites u nder p olygonal n orm s in the p lane, obtaining results for the E uclidean case thr ough appr o ximation argumen ts. F or u s, a crucial obser v ation is that the uniqueness of zone diagrams do es not hold for normed spaces. Let us consider R 2 with the ℓ 1 norm k·k 1 , giv en by k x k 1 = | x 1 | + | x 2 | . It is easy to c h ec k that the tw o p oin t sites (0 , 0) and (0 , 3) hav e at least t wo differen t zone diagrams, as d ra wn in Fig. 2. This example w as essen tially con tained already in Asano and Kirkp atric k [1 ], although in a different cont ext. The ℓ 1 norm d iffers from t he Eu clidean norm in t w o basic resp ects: the unit ball h as sharp corners and straight edges; in other wo rds, the ℓ 1 norm is neither smo oth nor r otund. W e recall that a norm k·k on R d is called smo oth if th e fun ction x 7→ k x k is different iable (geometricall y , the u nit ball of a sm o oth n orm has no “sharp corners”; s ee Fig. 3). 4 A n orm k·k on R d is called r otund (or strictly c onvex ) if f or all x, y ∈ R d with k x k = k y k = 1 and x 6 = y we hav e k x + y 2 k < 1. Geometrical ly , the un it sph ere of k·k con tains no segmen t. By compactness, a rotund norm on a finite-dimensional sp ace is a lso uniformly c onvex , whic h means that for ev ery ε > 0 there is µ = µ ( ε ) > 0 suc h that if x, y are unit v ectors with k x − y k ≥ ε , then     x + y 2     ≤ 1 − µ (w e r efer to [5] for this and other facts on n orms mentio ned without pro ofs). The Eu clidean n orm k·k 2 , and more generally , the ℓ p norms with 1 < p < ∞ , are b oth rotund and smo oth. W e hav e the follo wing g eneralization of T heorem 1.1: Theorem 1.2. L et the c onsider e d metric sp ac e ( X , dist) b e R d with a norm k·k that is b oth smo oth and r otund. F or every n -tuple P = ( P 1 , . . . , P n ) of nonempty close d sites in R d such that d ist( P i , P j ) > 0 for every i 6 = j , ther e exists exactly one zone diagr am R . 3 A fin ite- dimensional (real) normed space can be though t of a s t he real v ector space R d with some norm , whic h is a mapping that assigns a nonnegative real num b er k x k to eac h x ∈ R d so that k x k = 0 implies x = 0, k αx k = | α | · k x k for all α ∈ R , and the triangle inequality h olds: k x + y k ≤ k x k + k y k . Eac h norm k·k defi nes a metric by d ist( x, y ) := k x − y k . F or study ing a norm k·k , it is usually goo d to lo ok a t its uni t b al l { x ∈ R d : k x k ≤ 1 } . The un it ball of any norm is a closed conv ex b o dy K t h at is symmetric ab out 0 and contains 0 in the interior. Conv ersely , any K ⊂ R d with t h e listed prop erties is the unit ball of a (uniq u ely determined) norm. 4 There are sev eral n otions of differentiabili ty of functions on Banac h spaces , such as t h e existence of directional deriv atives, Gˆ ateaux differentia bility , F r´ echet differentiabil ity , or uniform F r´ ec het differentiabilit y . How ever, in finite-dimensional Banac h spaces they are all equiv alent. KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 3 rotund but not smo oth smo oth but not rotund smooth and rotund Figure 3. Rotundit y and smo othness of n orms. The pr o of for the Euclidean case, i.e., of T heorem 1.1, is set u p so that it generalizes to smo oth and rotund norms more or less immediately; ther e is only one lemma wh ere we n eed to w ork h arder—see S ection 4. Our cu r ren t p ro of metho d app aren tly dep end s b oth on smo othn ess and on rotund it y . In Section 5 we show that smo othness is indeed essen tial, b y exhibiting a non-smo oth but rotund norm in R d with non-uniqu e zone d iagrams. On th e other hand, we s usp ect that the assumption of rotun dit y in Theorem 1.2 can b e dropp ed. Currentl y we ha v e a pr o of (see App endix A) only in a rather sp ecial case: Theorem 1.3. F or two p oint sites P 0 = { p 0 } and P 1 = { p 1 } in the plane R 2 with a smo oth norm, ther e exists exactly o ne zo ne d iagr am. 2. Preliminaries Here w e introdu ce n otation and presen t some results from the literature, some of them in a more general cont ext than in th e original pu blications. Let ( X , dist) b e a general metric space. The closure of a set A ⊆ X is denote d by A , while ∂ A stand s f or its b oundary . The (closed) b all of radius r cente red at x is denoted by B( x, r ). F or sets A, B ⊆ X , not b oth empt y , we d efine the dominanc e r e gion of A o ver B as the set dom( A, B ) := { x ∈ X : dist( x, A ) ≤ dist( x, B ) } , where dist( C, D ) := inf x ∈ C, y ∈ D dist( x, y ) ∈ [0 , + ∞ ] denotes the distance of sets C and D . Let us fi x an n -tuple P = ( P 1 , . . . , P n ) of sites, i.e., nonempt y s u bsets of X (which, as ab ov e, w e assume to b e disjoin t and closed). F or an n -tuple R = ( R 1 , . . . , R n ) of arbitrary su bsets of X , we define an other n -tuple of regions R ′ = ( R ′ 1 , . . . , R ′ n ) denoted by Dom R and giv en by R ′ i := dom  P i , [ j 6 = i R j  , i = 1 , . . . , n (the sites are considered fixed and they are a part of the definition of the op erator Dom ). The defin ition of a zone diagram can no w b e expressed as follo ws: An n -tup le R is called a zone diagr am for the n -tuple P of sit es if R = Dom R (comp onent wise equalit y , i.e., R i = dom  P i , S j 6 = i R j  for all i ). F or t w o n -tuples R and S of sets, w e write R  S if R i ⊆ S i for ev ery i . It is easily seen (see, e.g., [2]) that the op erator Dom is an timonotone, i.e., R  S implies Dom R  Dom S . Our s tarting p oint in the pr o of of Theorems 1.1 and 1.2 is the follo w ing general result (see App end ix B f or a pro of ): Theorem 2.1 ( [2, Lemma 5.1], [8, T heorem 5.5]) . F or every n - tuple P o f sites (in any metric sp ac e) ther e exist n - tu ples R and S such that R = Dom S and S = Dom R . Mor e over, for every n -tuples R ′ , S ′ with R ′ = Dom S ′ and S ′ = Dom R ′ we have R  R ′ , S ′  S (and in p articular, R  S ). W e finish this section with a simp le geometric lemma. It w as used, in a less general setting, in [2 ] (pr o of of Lemma 4.3). 4 ZONE DIAGRAMS IN N ORMED SP ACES P i p a K ε 4 Figure 4. The cone K . P 1 P 3 P 2 R 3 R 1 = S 1 R 2 S 3 S 2 p a b δ s Figure 5. The setting of the pr o of of Theorem 1.1 (a schemati c picture). Observ ation 2.2. L et P b e an n -tuple of sites (in an arbitr ary metric sp ac e), and supp ose that ε := min i 6 = j dist( P i , P j ) > 0 and that R and S satisfy R = Dom S and S = Dom R . Then dist( P i , S j 6 = i S j ) ≥ ε 2 , and c onse que ntly, the ε 4 -neighb orho o d of e ach P i is c ontaine d in R i . Pr o of. W e recall the s im p le pro of from [2]. W e first note that V = ( V 1 , . . . , V n ) := Dom P is the classical V oronoi diagram of P , and the op en ε 2 -neigh b orho o d of P i do es not intersect S j 6 = i V j . Sin ce P  R , we h a ve Dom P  Dom R = S , and h en ce the op en ε 2 -neigh b orho o d of P i is d isj oin t f rom S j 6 = i S j as well , as claimed.  3. The Euclidean case Here we pr ov e Theorem 1.1; thr ou gh ou t this section, d ist denotes the Euclidean d istance. In addition to T h eorem 2.1 and Observ ation 2.2 , we also need the n ext lemma. Lemma 3.1 (Cone lemma, Euclidean case) . L et P b e an n -tuple of (nonempty close d) sites in R d with the Euclide an metric with ε := min i 6 = j dist( P i , P j ) > 0 , and let R a nd S satisfy R = Dom S and S = Dom R . L et a b e a p oint of some R i , and let p ∈ P i b e a p oint of the c orr esp onding site closest to a (such a ne ar est p oint exists by c omp actness). Then the set K := conv  { a } ∪ B( p, ε 4 )  is c ontaine d in R i ; se e Fig. 4. The follo wing pro of is rather sp ecific for the Eu clidean metric (the lemma f ails f or the ℓ 1 metric, f or example). Pr o of. Both a and B( p, ε 4 ) are cont ained in dom( p, S j 6 = i S j ) (the latter by Observ ation 2.2). F or the Eu clidean metric, the d ominance region of a p oin t o v er an y set is con vex, since it is the in tersection of halfspaces. Hence K ⊆ dom( p, S j 6 = i S j ) ⊆ R i .  No w we describ e the general strategy of the pr o of of Theorem 1.1. With R and S as in Theorem 2.1 , it suffices to pro ve R = S . F or c on tradiction, w e assu me that it is not the case, i.e., that R := S n i =1 R i is p rop erly cont ained in S := S n i =1 S i ; see the schemati c illustration in Fig. 5. F or a p oint b ∈ S \ P , let s ( b ) := dist( b, P ) b e the distance fr om the nearest site, and let p = p ( b ) ∈ P i b e a p oin t where this distance is attained. Let a = a ( b ) b e the closest p oint to b that lies in the intersecti on of R i with the segment bp . It is easily seen, using the triangle inequalit y , that p is also a nearest p oin t o f P to a . Thus, the set K in Lemma 3.1 is KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 5 p a b b ′ r r δ s ≥ s δ ′ a ′ p ′ s ′ ≤ dist( b ′ , R i ) Figure 6. The construction of b ′ . con tained in R i , and in partic ular, a is th e only in tersection of the segment bp with ∂ R i . W e set δ ( b ) : = dist( b, a ). T he paramet ers s ( b ) and δ ( b ) will measure, in some se nse, ho w m u c h S differs from R “at b ”. Assuming R 6 = S , we c ho ose a p oint b 0 ∈ S \ R . Then, using b 0 , w e fi nd b 1 ∈ S \ R where S differs from R “more than” at b 1 . Iterating the same pro cedur e we obtain an infi nite sequence b 0 , b 1 , b 2 , b 3 , . . . of p oints, and the d ifference will “gro w ” b ey ond b ounds, wh ile, on the other hand, it has to sta y b ounded—and this wa y w e r eac h a con tradiction. More concrete ly , for eve ry integ er t ≥ 1 w e will construct b t from b t − 1 so that, with s := s ( b t − 1 ), s ′ := s ( b t ), δ := δ ( b t − 1 ), and δ ′ := δ ( b t ), w e hav e (A) s ′ ≤ s − α , or (B) s ′ ≤ s − δ and δ ′ ≥ δ , where α > 0 is a constan t that d ep ends on s 0 := s ( b 0 ) and ε , but not on t . Th us, as t increases, s ( b t ) ke eps decreasing. S in ce s ( b t ) is b oun ded from b elo w by ε 4 b y Observ ation 2.2, case (A) can happ en on ly fi nitely many times. T herefore, fr om some t on, w e ha v e case (B) only . But this also causes s ( b t ) to decrease to wards 0—a con tradiction. It remains to describ e the construction of b t from b t − 1 , and this is done in the next lemma. Lemma 3.2. F or ev ery s 0 and ε > 0 ther e exists α > 0 such that if b ∈ S \ R satisfies s := s ( b ) ≤ s 0 , then ther e exists another p oint b ′ ∈ S \ R such that s ′ := s ( b ′ ) , δ := δ ( b ) and δ ′ := δ ( b ′ ) satisfy (A) or (B) . Pr o of. Let b ∈ S i , let a := a ( b ), p := p ( b ), and write r = dist( a, p ); see Fig. 6. Since a ∈ ∂ R i and R = Dom S , there exist j 6 = i and b ′ ∈ S j with d ist( a, b ′ ) = r . If th ere are sev eral p ossib le b ′ , w e choose one of them arb itrarily . First w e c hec k that b ′ 6∈ R , or in other w ords, that δ ′ > 0. Dur ing this step we also deriv e a lo w er b ound for δ ′ that will b e useful later. Since b ∈ S , a ′ ∈ R , a nd S = Dom R , we ha v e dist( a ′ , b ) ≥ s . Then we b ound, using the triangle in equ alit y , (1) δ ′ ≥ dist( a ′ , b ) − dist( b, b ′ ) ≥ s − dist( b, b ′ ) . Supp osing for con tradiction that δ ′ = 0, w e get dist( b, b ′ ) = s . But the triangle inequ alit y giv es dist( b, b ′ ) ≤ dist( b, a ) + d ist( a, b ′ ) = r + δ = s , and h ence the triangle inequ alit y here holds with equalit y . F or the Euclidean metric, this can happ en only if a lies o n the segmen t bb ′ , and then b ′ has to coincide with p , whic h is imp ossib le. So δ ′ > 0 indeed. Next, s in ce S = Dom R and b ′ ∈ S , w e ha ve s ′ ≤ dist( b ′ , R i ). An obvio us upp er b ound for dist( b ′ , R i ) is dist( b ′ , a ) = r = s − δ , and thus the first inequalit y in (B), n amely , s ′ ≤ s − δ , alw ays holds. Moreo ve r, if δ ≥ α , then s ′ ≤ s − δ ≤ s − α , and w e ha ve (A). F or the rest of the pro of w e thus assume that δ < α (where α hasn’t b een fi xed ye t—so far w e’re free to c h o ose it as a p ositiv e function of ε and s 0 in any wa y we lik e). Let us co nsider the ball B( b ′ , r ); see Fig. 7. If it con tains b , as in the left picture, we ha v e dist( b ′ , b ) ≤ r , and thus b y (1) w e hav e δ ′ ≥ s − r = δ . Then (B) holds. Thus, the last case to deal with is b 6∈ B( b ′ , r ). 6 ZONE DIAGRAMS IN N ORMED SP ACES p a b b ′ r δ ≤ r p a b b ′ r K γ c c ′ ℓ k β Figure 7. The r -ball around b ′ . (0 , 0) (1 , 2) Figure 8. The domin an ce region of th e p oin t (0 , 0) against (2 , 1) in the ℓ 4 norm. Let us consid er the cone K = conv( { a } ∪ B( p, ε 4 )) as in Lemma 3.1. Its op ening angle γ is b ound ed a wa y f r om 0 in terms of ε and s 0 . Let Π b e a 2-dimensional p lane con taining p, a, b ′ ; it also con tains b since p, a, b are collinear. Let k b e the ra y originating at a and con taining b , and let ℓ b e the ra y in Π orig inating at a and making the angle π − γ 2 with k (on th e side of b ′ ); see Fig. 7 righ t. Since th e angle of the ra ys k and ℓ is b oun ded a w a y fr om the straig ht angle, the Euclidean ball B( b ′ , r ) cuts a segmen t of significan t length β f rom a t least one of these rays; here β can b e b ounded from b elo w b y a p ositiv e quan tit y dep ending only on s 0 and ε . So far w e ha v en’t fixed α , and so now w e ca n mak e sure th at α < β . Since w e a ssume b 6∈ B( b ′ , r ), the segmen t of length β cut out by B( b ′ , r ) can’t b elong to the ra y k . So th e situation is as in Fig. 7 righ t: B( b ′ , r ) con tains the initial segmen t ac of ℓ of length β . Hence dist( b ′ , c ) ≤ r . The distance dist( c, R d \ K ) is b ound ed a w a y from 0 in terms of β and γ , and so w e may fix α so that dist( c, R d \ K ) ≥ α . Let c ′ b e the p oin t w h ere the segment b ′ c meets the b oundary of K . W e ha ve dist( b ′ , K ) ≤ d ist( b ′ , c ′ ) = dist( b ′ , c ) − dist( c, c ′ ) ≤ r − dist( c, R d \ K ) ≤ r − α. Then, finally , using K ⊆ R i , we hav e s ′ ≤ dist( b ′ , R i ) ≤ dist( b ′ , K ) ≤ r − α < s − α, and so (A) holds. This conclud es the pro of of Lemma 3.2, as well as that of Theorem 1.1.  4. The case of smoo th and rotund norms In this section we establish Theorem 1.2. W e b egin with the part wh ere the pro of differs from the Euclidean case: the cone lemma. In the Euclidean case, we used the fact that for p oin ts p 6 = q , dom( p, q ) is a halfspace, and consequen tly , d om( p, X ) is con vex f or arb itrary X . F or other norms dom( p, q ) need not b e conv ex, t hough; see Fig. 8. W e h a ve at least the follo wing conv exit y r esult. Lemma 4.1. L et us c onsider R d with an arbitr ary norm k·k , let H b e a close d halfsp ac e, and let p / ∈ H b e a p oint. Then dom( p, H ) is c onvex. KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 7 x y z H x ∗ y ∗ z ∗ p Figure 9. The domin an ce region of a p oin t against a h alfspace. p a B ( a, r ) B ( p, ε 2 ) Figure 10. T he sets C (shaded) and D . Conse quently, if the c omplement of a close d set A ⊆ R d is c onvex and p / ∈ A , then dom( p, A ) is c onvex. Pr o of. Let x / ∈ H b e a p oint and let x ∗ ∈ ∂ H b e a p oin t wh ere dist( x, H ), the distance of x to H measured by k·k , is attained. If y 6∈ H is anot her p oin t and y ∗ ∈ ∂ H is the p oin t suc h that the v ectors x − x ∗ and y − y ∗ are parallel, th en k y − y ∗ k = d ist( y , H ); see Fig. 9. No w let x, y ∈ dom( p, H ), let x ∗ , y ∗ b e as ab ov e, set z := ( x + y ) / 2, and let z ∗ b e defined analogously to y ∗ . Then we get dist( z , H ) = k z − z ∗ k = ( k x − x ∗ k + k y − y ∗ k ) / 2 = (dist( x, H ) + dist( y , H )) / 2. F rom this z ∈ dom( p, H ) is immediate, since k p − z k ≤ ( k p − x k + k p − y k ) / 2 ≤ (dist( x, H ) + dist( y , H )) / 2 = dist( z , H ). This pr o ves the fir s t part of the lemma. The second part follo ws easily: A can b e expressed as a union of c losed halfspaces H , and dom( p, A ) is the in tersection of the conv ex sets dom( p, H ).  No w we pr o ve a cone lemma, similar to Lemma 3.1: Lemma 4.2 (C on e lemma for rotund norms) . L et k·k b e a r otund norm on R d . Supp ose tha t an n - tuple P of sites satisfies ε := min i 6 = j dist( P i , P j ) > 0 , and R and S satisfy R = Dom S and S = Dom R . Then for every s 0 > 0 th er e is ρ > 0 (also d ep ending on ε and o n k·k ) suc h that the fol lowing hold s: If a ∈ R i with r := dist( a, P i ) ≤ s 0 and p ∈ P i is a p oint attaining the distanc e dist( a, P i ) , then the set K := conv  { a } ∪ B( p, ρ )  is c ontaine d in R i . Pr o of. As in th e Euclidean case, we b egin b y observing that a ∈ d om( p, S j 6 = i S j ) an d also B( p, ε 4 ) ⊆ dom( p, S j 6 = i S j ) by Observ ation 2.2. Thus, the set D := B( a, r ) ∪ B( p, ε 2 ) is con tained in the closure of R d \ S j 6 = i S j . W e no w w an t to fin d a op en con vex subset C ⊆ D su ch that a and B ( p, ρ ) are cont ained in dom( p, R d \ C ), since the latter regio n is co n v ex b y Lemma 4 .1 and th u s it cont ains K as well. W e let C b e the interio r of con v( B ( a, r ) ∪ B ( p, 2 ρ )) with ρ sufficien tly small (the restrictions on it will b e app aren t from the pr o of b elow); s ee Fig. 10. It is clear that { a } ∪ B ( p, ρ ) ⊆ dom( p, R d \ C ), and so it remains to pr o ve C ⊆ D . T o this end , it is s ufficien t to pro ve the foll o wing: If B := B(0 , 1) is the unit ball of k ·k and η > 0 is given, then there exists δ > 0 such th at for ev ery x ∈ R d with k x k ≤ 1 + δ , the “cap” 8 ZONE DIAGRAMS IN N ORMED SP ACES Figure 11. A schematic illustration of the u nit b all of k·k (1) . con v ( B ∪ { x } ) \ B has diameter at most η . This is a wel l-kno wn and easily prov ed pr op erty of u niformly conv ex norms. (Pro of sk etc h: If x with k x k = 1 + δ has a cap o f large diameter, then there is z of norm 1 and half of the diameter a w a y fr om x suc h that the line xz a v oids the in terior of B . L et y b e the other in tersection of this line with ∂ B(0 , 1 + δ )—then xy is a long segmen t that cuts in B(0 , 1 + δ ) into depth only δ .)  Pr o of of The or em 1 .2. The o v erall strategy of the p ro of is exactl y as for Theorem 1.1 (see S ec- tion 3). Th e constant α in (A) ma y also dep end on the considered norm k·k . This quantificat ion also n eeds to b e added in th e appr opriate version of Lemma 3.2. In the p ro of of that lemma, the first place where w e use a p r op erty n ot shared b y all n orms is b elo w (1); we need that the triangle in equalit y ma y hold with equalit y only for collinear p oints—this remains tr ue for all r otund norms. Then we pro ceed as in the Euclidean case, introdu cing the the cone K = conv( { a } ∪ B ( p, ρ )) as in Lemma 4.2. There is some γ > 0, dep endin g o n ε , s 0 , and the norm k·k , suc h that the appropriate Euclidean cone with op en ing angle γ is cont ained in K . (Here and in the s equ el we implicitly use the f act th at every n orm on R d is b et w een tw o constan t m ultiples of the Eu clidean norm, whic h is w ell kno wn and immediate by compactness.) W e define the ra ys k and ℓ , again f ollo wing t he Euclidean pro of. F or the next step, we need that, since the angle of these ra ys is b oun ded a w a y from the straigh t an gle, at least one of k , ℓ cuts a segmen t of a significan t length β from the ball B( b ′ , r ). It is easy to see that this prop ert y follo ws fr om the smo othness of the norm. The rest of the pro of goes through unc hanged.  5. Non-uniquenes s examples As we sa w in the in tro duction, t w o p oin t s ites w ith the same x -co ordinate hav e at least tw o zone diagrams un der th e ℓ 1 metric. Here we show that only the n on-smo othness (sharp corners) of the ℓ 1 unit ball is essen tial for this example, while th e straigh t edges can b e replaced by curv ed ones. Prop osition 5.1. Ther e exists a r otund norm in the plane, arbitr arily close to the ℓ 1 norm, such that t wo distinct p oint sites with th e same x -c o or dinate have (at le ast) two differ ent zone diagr ams. The appropriate norm is n ot difficult to describ e, but proving the non-u n iqueness of the zone diagram is more d emanding, sin ce it seems hard to find an explicit description of a zone diagram for n on-p olygonal norms. Informally , w e constr u ct the desired norm b y sligh tly “inflating” the unit ball of the planar ℓ 1 norm, so that the edges bulge out and the norm b ecomes rotund . It is imp ortan t t hat the inflation is asymmetric, as is sc hematically indicated in Fig. 11 (in the “real” example we infl ate m uc h less). W e will denote the resulting norm by k·k (1) ; th e su bscript should remind of “infl ated ℓ 1 ” graphically . KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 9 p q p q Figure 12. T he b isector of p and q under the ℓ 1 norm and u nder k·k (1) (sc h ematic). ε p q slop e ε bisector go es here Figure 13. T he cond itions in Lemma 5.2. T o explain the purp ose of the asymmetry in our example, w e consider the bisector of the p oints p = ( − 1 , 1) and q = (1 , − 1), i.e., the set of all p oin ts equidistant to p and q . F or the ℓ 1 norm, the b isector is “fat”, as sho w n in Fig. 12 left—it consists of a segment and t w o quadrants. By a small infl ation, whic h m akes th e norm rotund, the m id dle segment of th e bisector is c h anged only v ery sligh tly , b ut the “am biguit y” of the ℓ 1 bisector in the quadr an ts is “resolv ed”, and the quadran ts collapse to (p ossibly curv ed) ra ys. No w if the inflation w ere symmetric, we would get straight rays with slop e 1 in the bisector, bu t with an asymmetric inflation, w e can get a (p ositiv e) slo p e as small as w e wish. In order to establish the requ ired prop erties of the bisector formally , a safe r oute (if p erhaps not the most conceptual one) is to describ e k·k (1) analyticall y . The r a ys of the bisector will b e sligh tly cur v ed r ather than straigh t, bu t for the zone diagram construction th is will do as w ell. Lemma 5.2. F or every ε > 0 ther e exists a r otund norm k·k (1) in the pla ne, whose unit b al l c ontains the ℓ 1 unit b al l and is c ontaine d in the o ctagon as i n Fig. 13 left, such that the p ortion of the bise ctor of the p oints p = ( − 1 , 1) and q = (1 , − 1) lying i n the quadr ant { ( x, y ) : x, y ≥ 1 } is an x -monoto ne curve lying b e low the line y = ε ( x − 1) + 1 (Fig. 13 r ight). See App endix C for a pr o of. Pr o of of Pr op osition 5.1. W e sh o w th at the zone diagram of the sites p − = (0 , − 1) and p + = (0 , +1) u nder th e norm k·k (1) as in the lemma, with ε sufficiently small, is not un ique. First we consider the zone d iagram only inside the v ertical strip V := { ( x, y ) ∈ R 2 : x ∈ [ − 2 , 2] } . Let R + 0 b e the regio n as in Fig. 14, i.e., the part of t he region of p − within V in an ℓ 1 zone diagram of p − , p + . Let S + 0 b e obtained b y pulling the b ottom verte x of R + 0 do wn w ard b y η (whic h is another small p ositiv e parameter), and let R − 0 , S − 0 b e the reflections of R + 0 , S + 0 b y the x -axis. Let us consider the region dom( p − , R + 0 ) inside V (distances measured by o ur norm k·k (1) ). F or ev ery p oin t x ∈ V b elo w R + 0 , the k·k (1) -distance to R + 0 coincides with the ℓ 1 distance, 10 ZONE DIAGRAMS IN N ORMED SP ACES R + 0 R − 0 S + 0 S − 0 p + p − Figure 14. T he r egions R + 0 , S + 0 , R − 0 , S − 0 in th e ve rtical strip V . S + R − ˜ S + (2 , − 1) p − p + Figure 15. T he r egion ˜ S + defined u sing b isectors, and a region conta ining ˜ R − . whic h is simply the length of the vertica l segmen t from x to ∂ R + 0 . F rom this it is clear that dom( p − , R + 0 ) ⊇ R − 0 (since R − 0 is the d omin ance region of p − against R + 0 in the ℓ 1 metric, a nd k·k (1) ≤ k·k 1 ). Moreo ve r, it’s easy to c hec k that for ε (the parameter con trolling the c hoice of k·k (1) ) sufficien tly small, we also ha v e dom( p − , R + 0 ) ⊆ S − 0 . Th us, we ha ve R − 0 ⊆ dom ( p − , R + 0 ) ⊆ S − 0 , and by the vertica l symmetry w e also get R + 0 ⊆ dom( p + , R − 0 ) ⊆ S + 0 . Arguing as in either of the pro ofs of Th eorem 2.1, we get that there e xist regions R + , R − , S + , S − , wh ere R − is the reflection of R + , S − is the r eflection of S + , su ch th at R − 0 ⊆ R + ⊆ S + ⊆ S + , and ( R − , S + ) is a zone diagram of ( p − , p + ) (and so is ( S − , R + ), but we actually ha ve R + = S + , although we will n either need this nor pro v e it). All of this refers to the vertica l s tr ip V (so , formally , the metric space in th ese argum en ts is V with the k·k (1) metric). No w w e mo v e on to the full plane R 2 , and w e let ˜ S + b e the region consisting of S + plus t w o p arts of the upp er halfplane o utside V as in Fig. 15: Th e righ t part is delimited b y a part of the bisector of p + and (2 , − 1) (d ra wn thic k), and the left part b y a part of the bisector of p + and ( − 2 , − 1). No w we set ˜ R − := d om( p − , ˜ S + ). The distance of p oints inside V \ S + to ˜ S + is still the v ertical distance, i.e., the same as the distance to S + , and s o ˜ R − ∩ V = R − . F or th e part of ˜ R − outside V , we don’t need an exact descrip tion—it is s u fficien t that it lies b elo w the d ashed ra ys in Fig. 15 (using the p rop erty of the bisectors as in Lemma 5.2, one can see that these rays can b e tak en as steep as desired, by setting ε s u fficien tly small). F rom th is we can see th at for ev er y p oint of the upp er halfplane on the righ t of V , the nearest p oint of ˜ R − is the corner (2 , − 1). Therefore, d om( p + , ˜ R − ) = ˜ S + , and h ence ( ˜ R − , ˜ S + ) is a zone diagram of ( p − , p + ). But the mirror reflection of this zone diagram ab out the x -axis yields another, differen t zone dia- gram.  KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 11 Ac knowledgemen ts. W e are grateful to T etsuo Asano for v aluable discussions includin g those on n on-uniqueness exa mples for con v ex p olygonal dista nces. W e also expr ess our gratitude to Daniel Reem for careful reading and useful suggestions on th e man uscript. Finally , w e remark that the warm comment s from the aud ience of our preliminary annou n cemen t of p artial r esu lts at Eu roCG 2009 encouraged us to work furth er. Referen ces [1] T. Asano and D. Kirkp atric k. Distance trisector cu rves in regular conv ex dis tance metrics. In Pr o c. 3r d International Symp osium on V or onoi Diagr ams in Scienc e and Engine ering , IEEE Computer So ciety , p ages 8–17, 2006. [2] T. Asano, J. Matou ˇ sek, and T. T okuyama. Zone diagrams: Existence, uniqueness, and algo rithmic challenge. SIAM Journal on C om puting , 37(4):1182–119 8, 2007. [3] T. Asano, J . Matou ˇ sek, and T. T oku yama. The d istance trisector curve. A dvanc es i n Mathematics , 212(1):338 –360, 2007. [4] F. Aurenh ammer. V oronoi diagrams—a survey of a fundamental geometric data structure. ACM Computing Surveys , 23(3):345– 405, 1991. [5] Y. Benya mini and J. Lindenstrauss. Nonline ar F unctional Analysis, V ol. I, Col lo quium Public ations 48 . American Mathematical So ciety (A MS), Providence, RI, 1999. [6] J. Ch un, Y. Ok ada, and T. T okuyama. Distance trisector of segme nts and zone diagram of segments in a plane. In Pr o c. 4th International Symp osium on V or onoi Diagr ams in Scienc e and Engine ering , IEEE Computer Society , pages 66–73, 2007. [7] A. Okabe, B. Boots, K. Su gihara, and S. N. Chiu. Sp atial T essel lations: Conc epts and Applic ations of Vor onoi Diagr ams . Probability and Statistics. Wiley , second edition, 2000. [8] D. Reem and S. Reich. Zone and double zone diagrams in abstract spaces. Col lo quium Mathematicum 115(1):129 –145, 2009. [9] A. T arski. A lattice-theoretical fixp oint theorem and its applications. Pacific Journal of Mathematics , 5:285– 309, 1955. Appendix A. Pr oof of The o rem 1.3 Prop osition 5.1 sho wed that the assumption of smo othn ess in Theorem 1.2 cannot b e dropp ed, ev en for the simplest case of tw o singleton sites in the plane. T heorem 1.3, whic h we will prov e here, states that the rotundity assumption can b e dropp ed in this sp ecial case. Smo othness of th e norm means th at a metric ball h as a un ique supp orting halfspace at ev ery p oint in its su rface. T h us, for a nonzero v ector a , we can defi n e ⊤ > 0 a to b e th e open halfsp ace that touc hes (but not in tersects) the ball B( − a, k a k ) a t the origin. W e wr ite ⊤ ≤ 0 a = R d \ ⊤ > 0 a and ⊤ ≥ 0 a = ⊤ ≤ 0 − a . F or n on zero ve ctors a and b , define a ∼ b when ⊤ > 0 a = ⊤ > 0 b . T hen ∼ is an equiv alence relation. It is easy to see (Fig. 16) that for nonzero ve ctors a 1 , . . . , a m , w e h a ve (2) k a 1 + · · · + a m k = k a 1 k + · · · + k a m k if and only if a 1 ∼ · · · ∼ a m . Lemma A.1. L et k·k b e a smo oth norm on R d . Then ther e ar e p ositive numb ers α and β such that for any unit ve ctors u , v with k u + v k > 2 − β , we have k u − αv k ≤ 1 . Pr o of. The angle σ u b et w een a un it vec tor u and ⊤ ≤ 0 u is a con tinuous function of u , and hence attains a p ositiv e m inim um σ . Let ⊤ ≥ σ/ 2 u (and ⊤ ≤ σ/ 2 u ) b e the set of vecto rs (includ ing 0) that mak e an angle ≥ σ / 2 (and ≤ σ / 2) with ⊤ ≤ 0 u (Fig. 17). W e find the desired α and β as follo ws. b a b a B( − b, k b k ) B( − b, k b k ) B( a, k a k ) B( a, k a k ) Figure 16. k a + b k = k a k + k b k if and only if a ∼ b (equation (2 ) with m = 2). 12 ZONE DIAGRAMS IN N ORMED SP ACES u ⊤ > 0 u unit ball σ u σ ⊤ ≥ σ / 2 u Figure 17. ⊤ ≥ σ/ 2 u is the set of v ectors that are significan tly closer to u than to − u . 0 η y z B( u, k u k ) B( v , k v k ) ⊤ ≥ 0 u ⊤ ≥ v Figure 18. Wh en u and v are close, y ∈ B( v , k v k ) is not v ery far fr om B( u, k u k ). F or un it ve ctors u and v w ith v ∈ ⊤ ≥ σ/ 2 u , let α u,v b e the length of the segmen t that the un it ball cuts out from the line u + R v . In other wo rds, α u,v is the unique p ositiv e n umber su c h that k u − α u,v v k = 1. Th en α u,v is con tinuous in u and v , and th u s attains a p ositiv e minim um α . F or unit v ectors u and v with v ∈ ⊤ ≤ σ/ 2 u , let β u,v = 2 − k u + v k . Then β u,v is p ositiv e and con tinuous in u and v , and th us attains a p ositiv e minimum β . Since ⊤ ≥ σ/ 2 and ⊤ ≤ σ/ 2 co vers the w hole space, α and β ha v e th e stated prop ert y .  Lemma A.2. L et k·k b e a smo oth norm on R d . F or any κ > 0 , ther e is ε > 0 such t hat, for any ve ctors u , v with k u k , k v k ≥ 1 and k u − v k < ε , we ha ve dist( y, B( u, k u k )) < κ k y k for any y ∈ B( v , k v k ) . Pr o of. Since dist( y , B( u, k u k )) ≤ 2 ε , it is clear that, for an y c onstan t η > 0, the claim holds if w e consider only those y with k y k ≥ η . Therefore, it suffices to prov e the existence of η > 0, dep end ing on k·k and κ , suc h that the claim holds f or any y with k y k < η . W e find the desired η and ε as follo ws (Fig. 18). Since the norm is smo oth, the su r face of a b all lo oks lik e a hyp erplane lo cally at eac h p oin t. Thus, there exists η > 0 such that f or any u ∈ R d with k u k ≥ 1 and any z ∈ ⊤ ≥ 0 u with k z k < η (1 + κ/ 2), we ha v e dist( z , B( u, k u k )) ≤ κ k z k / (2 + κ ). Also, since c hanging slightly a ve ctor u of length 1 or greater m ov es ⊤ ≥ 0 only slightly , there is ε > 0 so small that for any v ectors u , v of length 1 or greater with k u − v k < ε , w e h a ve dist( y , ⊤ ≥ 0 u ) ≤ κ k y k / (2 η ) for all y ∈ ⊤ ≥ 0 v . Since y ∈ B( v , k v k ) ⊆ ⊤ ≥ 0 v , we hav e dist( y, ⊤ ≥ 0 u ) ≤ κ k y k / 2 b y our c hoice o f ε . Let z ∈ ⊤ ≥ 0 u b e a p oin t attaining this distance. Since k z k ≤ k y k + k z − y k ≤ k y k + κ k y k / 2 = k y k (1 + κ/ 2) ≤ η (1 + κ/ 2), we h a v e dist( z , B( u, k u k )) ≤ κ k z k / (2 + κ ) ≤ κ k y k / 2 by our c hoice of η . These imply dist( y , B( u, k u k )) < κ k y k b y the triangle inequalit y .  Lemma A.3. L et k·k b e a smo oth norm on R 2 . F or unit v e ctors u and v with k u − v k < 2 , ther e is κ > 0 such that for a l l y ∈ dom( v , u ) \ B( v , 1) sufficiently c lose to th e origin (Fig. 19), dist( y , B( u, 1)) ≥ κ k y k . KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 13 u v w dom( v , u ) 0 Figure 19. The conclusion of Lemma A.3 states that dom( v , u ) and the b ound- ary of B( u, 1) “mak e a p ositiv e angle” at the origin. W e pro ve this b y sho wing that there is a cone (shaded) whose axis is the tangen t v ector w and which do es not o v erlap dom( v , u ). Pr o of. Because k u − v k < 2, the vect ors u and − v do n ot share the tangent. Th erefore, there is a (uniqu e) unit v ector w ∈ ⊤ ≥ 0 u ∩ ⊤ ≤ 0 u that heads out of B( v, 1). Since lim δ → 0 k u − δ w k − 1 δ = 0 , β := l im δ → 0 k v − δ w k − 1 δ > 0 , there exists δ 0 > 0 so s mall th at f or all p ositive δ < δ 0 , we hav e k u − δ w k − 1 δ < 1 3 β , k v − δ w k − 1 δ > 2 3 β , and hence k u − δ w k < k v − δ w k − β δ / 3. This imp lies that k u − x k < k v − x k for all x ∈ B( δw , β δ / 6). Thus, dom( v , u ) is disjoint from a co ne (e xcept at the orig in) whose v ertex is at the origin and axis is the ve ctor w (see Fig. 19). This implies wh at is stated.  No w we lo ok at the situation of Theorem 1.3. Let R = ( R 0 , R 1 ) and S = ( S 0 , S 1 ) b e pairs satisfying R  S and R = Dom S , S = Dom R (which exist by Th eorem 2.1). As b efore, it suffices to sho w that R = S . S upp ose otherwise. Then h = min { d ist( p 0 , S 0 \ R 0 ) , d ist( p 1 , S 1 \ R 1 ) } exists. Lemma A.4. In the ab ove setting, if a p oint c ∈ S 0 \ R 0 satisfies k c − p 0 k = h , then (a) k c − p 1 k = 2 h ; (b) ther e is a p oint c ′ ∈ S 1 \ R 1 satisfying k c ′ − c k = k c ′ − p 1 k = h . Pr o of. Note that c ∈ R 0 , sin ce otherwise S 0 \ R 0 in tersects a part of the segmen t cp 0 of p ositiv e length, con tradicting the minimality of h . There is a sequence ( x i ) i ∈ N of p oints in S 0 \ R 0 that con v erges to c . F or eac h i ∈ N , let y i ∈ S 1 b e a closest p oint to x i . Sin ce x i ∈ S 0 \ R 0 , w e ha v e k y i − x i k = dist( x i , S 1 ) < k p 0 − x i k and y i ∈ S 1 \ R 1 . The sequence ( y i ) i ∈ N has a subsequen ce ( y j i ) i ∈ N that con v er ges to a p oin t c ′ ∈ S 1 \ R 1 (Fig. 20). Note that k c ′ − p 1 k ≤ k c − c ′ k = lim i →∞ k x j i − y j i k ≤ lim i →∞ k p 0 − x j i k = k p 0 − c k = h, where the first inequalit y is by c ′ ∈ S 1 and c ∈ R 0 . In fact, this holds in equalit y by the minimalit y of h . W e ha v e pr o ved (b). F or eac h i , s in ce S 1 \ R 1 in tersects a p art of the segment y j i c ′ of p ositiv e length, y j i / ∈ B( p 1 , h ) b y the m inimalit y of h . Also, y j i ∈ S 1 ⊆ dom( p 1 , c ). As i increa ses, y j i comes arbitrarily close to c ′ . Hence, if (a) is not tru e, Lemma A.3 giv es a constan t κ > 0 suc h that dist( y j i , B( c, h )) ≥ κ k y j i − c ′ k for all but fi nitely many i . On the other hand, since y j i is in B( x j i , k x j i − c ′ k ) and ( x j i ) i ∈ N con verges to c , Lemma A.2 sho ws th at dist( y j i , B( c, h )) < κ k y j i − c ′ k for all bu t fi nitely man y i . Th is is a con tr ad iction. W e ha v e pro v ed (a).  Lemma A.5. In the ab ove setting, k p 0 − p 1 k = 3 h . 14 ZONE DIAGRAMS IN N ORMED SP ACES p 1 p 0 p 0 c c ′ x 1 x 2 y j 1 y j 2 h h h Figure 20. Lemma A.4. Pr o of. By the definition of h , there is a p oint c ∈ S 0 \ R 0 satisfying k c − p 0 k = h . By Lemma A.4(b), there is a p oin t c ′ ∈ S 1 \ R 1 satisfying k c ′ − c k = k c ′ − p 1 k = h . By Lemma A.4(a) (and the same lemma with the sites sw ap p ed), k c − p 1 k = k c ′ − p 0 k = 2 h . This implies ( c − p 0 ) ∼ ( c ′ − c ) ∼ ( p 1 − c ′ ) and thus k p 0 − p 1 k = 3 h by (2) at the b eginnin g of this section.  T o p ro v e Theorem 1.3, w e will construct a sequence ( b t ) t ∈ N of p oin ts in R \ S , as w e d id in Section 3. Recall that for eac h i ∈ { 0 , 1 } and b ∈ S i , we defin e a ( b ) to b e the closest p oint to b th at is in the in tersection of R i with the segmen t bp i (note that since w e do not ha ve the cone lemma this time, the inte rsection of bp i and ∂ R i is not alw a ys unique). As b efore, let s ( b ) = k b − p i k and δ ( b ) = k b − a ( b ) k . The pr o of goes as follo ws. This time, we b egin w ith a p oin t b 0 ∈ S 0 \ R 0 that is within distance h + ε fr om the nearest site, for some small ε > 0 (suc h b 0 exists b y the definition of h ), and tak e b 1 , b 2 , . . . as w e did in Section 3 using Lemma 3.2: F or eac h b t ∈ S i \ R i , w e let b t +1 ∈ S 1 − i \ R 1 − i b e a p oint that is at the same distance from a ( b t ) as p i is. Th en eac h b t will b e also within dista nce h + ε from the nearest site p i . Because we hav e pr ov ed that the sites are 3 h apart, and the path p i - a ( b t )- b t +1 - p 1 − i consists of three segmen ts sh orter than h + ε , the path m ust b e “almost str aight” . This implies that we will alwa ys ha v e the case (B) in S ection 3 (Fig. 7 left): Lemma A.6 . In the ab ove setting, th e fol lowing holds for some ε > 0 : F or e ach i ∈ { 0 , 1 } and b ∈ S i \ R i satisfying s := s ( b ) < h + ε , ther e is b ′ ∈ S 1 − i \ R 1 − i such that δ := δ ( b ) , s ′ := s ( b ′ ) , δ ′ := δ ( b ′ ) satisfy (B) of Se ction 3 (i.e., δ ′ ≥ δ and s ′ ≤ s − δ ). Pr o of. Let ε := m in { hα, hβ / 3 } , where α and β are as in Lemma A.1. Let b b e as assumed. By the definition of a := a ( b ), there is b ′ ∈ S 1 − i with k b ′ − a k = k a − p i k . W e show t hat this b ′ qualifies. Since s ′ = k b ′ − p 1 − i k ≤ k b ′ − a k = k a − p 1 − i k = s − δ , it s u ffices to prov e that δ ′ ≥ δ (whic h would then imp ly b ′ / ∈ R 1 − i ). By Lemma A.5, w e ha ve k b ′ − p i k ≥ k p 1 − i − p i k − k p 1 − i − b ′ k = 3 h − s ′ > 3 h − s ≥ 3 h − ( h + ε ) = 2( h + ε ) − 3 ε ≥ 2( h + ε ) − β h > ( h + ε )(2 − β ) > k a − p i k (2 − β ) . By this and k b ′ − a k = k a − p i k , Lemma A.1 yields k ( b ′ − a ) − α ( a − p i ) k ≤ k a − p i k . This remains true if we decrease α , since B(0 , k a − p i k ) is con v ex. So w e replace α by k b − a k / k a − p i k ≤ ε/h ≤ α , obtaining k b ′ − b k = k ( b ′ − a ) − ( b − a ) k ≤ k a − p i k . Since b is in S i and a ′ := a ( b ′ ) is in R 1 − i , w e h a ve k a ′ − b k ≥ s . Hence, δ ′ = k b ′ − a ′ k ≥ k a ′ − b k − k b ′ − b k ≥ s − k a − p i k = δ , as desired.  KA W AMURA, MA TOU ˇ SEK, TOKUY AMA 15 The rest o f the argum en t is simila r to what we a lready sa w in Sec tion 3 (a nd ev en sim p ler b ecause we do not ha ve case (A) this time): S tarting at b 0 ∈ S \ R suc h that s ( b 0 ) < h + ε , where ε is as in Lemma A.6, we defi ne b t +1 , for eac h t ∈ N , to b e the p oint b ′ corresp ondin g to b = b t . By the lemma, s ( b t ) alwa ys decreases by at least δ ( b 0 ), leading to a cont radiction. Th is pro v es Theorem 1.3. Appendix B. Proofs of T heore m 2.1 There are t w o pro ofs of Theorem 2.1 av ailable; w e sket c h the main ideas for the reader’s con venience. The first pr o of , f r om [2], do esn’t establish the theorem in full g eneralit y— it w orks only for closed and disjoint sites in a Euclidean s pace, or more generally , in a finite-dimensional normed space with a rotund n orm. In this pro of, we build a sequence of inner approximat ions to R and outer app ro ximations t o S . Namely , we set R (0) := P , S (0) := Dom R (0) (this is the classical V oronoi diagram of the sites P 1 , . . . , P n ), and for k = 1 , 2 , . . . w e p ut R ( k ) := Dom S ( k − 1) , S ( k ) := Dom R ( k − 1) . An timonotonicit y of Dom and induction yield R (0)  R (1)  R (2)  · · · and S (0)  S (1)  S (2)  · · · , as well as R ( k )  S ( k ) for all k . W e then define R and S b y R i := ∞ [ k =0 R ( k ) i , S i := ∞ \ k =0 S ( k ) i . It remains to sh o w that R and S are as required. T his is d one in [2 ] for the case of p oint sites in R 2 with the Euclidean norm. By in s p ecting th e pro of (Lemma 5.1 of [2]), we see that it uses only the follo win g prop erty of the underlyin g metric space (stated there as Lemma 3.1): If P is a c lose d set, X 1 ⊇ X 2 ⊇ · · · is a de cr e asing se quenc e of close d sets with X 1 ∩ P = ∅ , and X := T ∞ k =1 X k , then dom( P , X ) ⊆ S ∞ k =1 dom( P, X k ) . ( Moreo ver, in the p ro of one also n eeds that P i ∩ S (0) j = ∅ for i 6 = j ; since we assume th e sites to b e closed and disjoin t, this pr op ert y of the V oronoi diagram is immediate.) T o verify the abov e state men t, w e c an ag ain f ollo w the p ro of of Lemma 3.1 in [2]. First w e c h ec k that with the X k as ab o ve and any p oin t y , w e ha v e dist( y , X ) = lim k →∞ dist( y , X k ); this follo ws easily assuming compactness of all closed balls in a fin ite-dimensional normed space. No w let us fix x ∈ dom( P , X ) arbitrarily (we ma y assume x 6∈ P , sin ce the case x ∈ P is clear) and choose ε > 0; w e wan t to sh o w that dist( x, dom ( P , X k )) ≤ ε for some k . W e let p b e a p oint of P n earest to x , a nd c ho ose a point y 6 = x o n the seg men t px at d istance smaller th an ε from x . It is easy to chec k, using the rotundit y of th e norm , that dist( y , p ) < dist( y , X ), and th us dist( y , p ) ≤ dist( y , X k ) for k sufficien tly large. S o y ∈ dom( P , X k ) and we are done. The se c ond pr o of of Th eorem 2.1, d ue to Reem and Reic h [8], is based on the follo wing theorem of K naster and T arski (see [9]): If L = ( L,  ) is a complete lattice and g : L → L is a monotone mapping, then g has at least one fixed p oin t (i.e., x ∈ L with g ( x ) = x ) , and moreo ver, th ere exists a smallest fixed p oint x 0 and a largest fixed p oin t x 1 , i.e., such that x 0  x  x 1 for ev ery fixed p oint x . T o prov e Th eorem 2.1, w e let L b e the system of all ordered n -tuples D suc h that P i ⊆ D i for ev ery i . W e introdu ce the orderin g  as ab o ve (one has to chec k th at this give s a complete lattice, whic h is straigh tforw ard). L et g := Dom 2 ; that is, g ( D ) := Dom ( Dom D ). Then we let R b e the smallest fixed p oint of g as in the Knaster– T arski theorem, and S := Dom R . Clearly Dom S = Dom 2 R = g ( R ) = R . Moreo v er, if R ′ , S ′ satisfy R ′ = Dom S ′ and S ′ = Dom R ′ , then R ′ and S ′ are b oth fixed p oin ts of Dom 2 , and th u s R  R ′ , S ′  S as claimed. Appendix C. Pr oof of Lemma 5 .2 The construction has tw o p ositiv e p arameters, α and δ , wh ere α is small and δ is still m uc h smaller. W e let k·k ′ b e the Euclidean norm scaled by α in th e horizon tal d irection; that is, k ( x, y ) k ′ = p α 2 x 2 + y 2 . Let k·k ′′ b e the ℓ 1 norm scaled b y a suitable factor β (close to 1) in the v ertical 16 ZONE DIAGRAMS IN N ORMED SP ACES direction: k ( x, y ) k ′′ = | x | + β | y | . The norm k·k (1) is obtained as a ′ k·k ′ + a ′′ k·k ′′ , wh ere a ′ , a ′′ > 0 are suitable co efficients. Th is obviously yields a norm, w hic h is rotund since k·k ′ is rotund. W e wa nt that th e contribution of k·k ′ is sm all compared to that of k·k ′′ , and that the corners of the unit ball of k·k (1) coincide with those of th e ℓ 1 unit ball. This fi nally leads to the formula k ( x, y ) k (1) := δ p α 2 x 2 + y 2 + (1 − αδ ) | x | + (1 − δ ) | y | . Fig. 11 is actually obtained from this formula with δ = 0 . 7 and α = 0 . 5. It is easy to c h ec k that, a s the p icture su ggests, k·k (1) ≤ k·k 1 (and thus the ℓ 1 unit ball is con tained in the k·k (1) unit ball), and for δ is sufficiently small in terms of α and ε , the un it ball of k·k (1) is con tained in th e octagon as in the lemma. It remains to inv estigate the bisector of p and q for x ≥ 1 and y ≥ 1. F or con v enience, w e translate p and q b y ( − 1 , − 1) and scale by 1 2 . Then the bisector is giv en b y the equation k ( x + 1 , y ) k (1) = k ( x, y + 1) k (1) , with the region of interest b eing the p ositiv e quad r an t x, y ≥ 0. F or x, y ≥ 0, th e absolute v alues can b e r emov ed, δ disapp ears fr om the equation, and we obtain p α 2 ( x + 1) 2 + y 2 + 1 − α = p α 2 x 2 + ( y + 1) 2 . This can b e solv ed for y explicitly , with the only p ositiv e ro ot y = 1 − α 2 − α  p 1 + 2 αx + 2 αx 2 − 1 + α 1 − α x  . This is the equ ation of the bisector curve in the p ositiv e q u adrant . It is a simple exercise in calculus (distinguishing th e cases αx ≤ 1 and αx > 1, say) to show that y ≤ C √ α x for all x > 0 and all sufficiently small α (here C is a suitable constan t).