The covariogram determines three-dimensional convex polytopes

THE CO V ARIOGRAM DETERMINE S THREE-DIMENSIONAL CONVEX POL YTOPES GABRIELE BIANCHI Abstract. The cross co v ariogram g K,L of tw o conv ex sets K , L ⊂ R n is the function which associates to eac h x ∈ R n the vo lume of the intersection of K with L + x . The problem of determining the sets from their co v ariogram is relev ant in stochastic geomet ry , in probabilit y and it is equiv alent to a partic- ular case of the phase retriev al problem in F ourier analysis. It is also relev an t for the inv ers e pr oblem of determining the atomic structure of a quasicrystal from its X- ray diﬀract ion im age. The tw o main results of this paper are that g K,K determines three-dimensional con vex polytop es K and that g K,L deter- mines both K and L when K and L are con v ex p olyhedral cones satisf ying certain assumptions. These results settle a conjecture of G. Matheron in the class of conv ex p olytopes. F urther r esults regard the known count erexamples in dimension n ≥ 4. W e also introduce and study the notion of synisothetic polytop es. Thi s concept is related to the rearrangemen t of the faces of a con ve x polytop e. 1. Introduction Let K be a conv ex b o dy in R n . The c ovario gr am g K of K is the function g K ( x ) = λ n ( K ∩ ( K + x )) , where x ∈ R n and λ n denotes n -dimensional Leb esgue meas ure. This functional, which w as introduced by Matheron in his b o ok [Mat7 5] on random sets, is also sometimes called the set c ovarianc e and it coincides with the au t o c orr elation of the characteristic function of K : (1.1) g K = 1 K ∗ 1 ( − K ) . The cov ariogr am g K is clearly unchanged by a tr a nslation or a re ﬂection o f K . (The term r eﬂe ction will a lwa ys mean reﬂection in a po int.) Mather on [Mat86] in 198 6 asked the following question and co njectured a p o sitive a nswer for the case n = 2. Co v ariogram problem. Do es the c ovario gr am determine a c onvex b o dy, among al l c onvex b o dies, up to tr anslations and r eﬂe ctions? The conjecture reg arding n = 2 has b een completely settled only v ery recently , by Averko v a nd Bianchi [AB]. Matheron [Mat75, p. 86 ] observed that, for u ∈ S n − 1 and for all r > 0, the deriv atives ( ∂ g K /∂ r )( r u ) g ive the dis tribution of the leng ths of the chords of K parallel to u . Suc h information is common in stereo logy , statistica l shap e r ecogni- tion a nd image ana lysis, when prop erties of a n unknown bo dy ar e to b e inferred from chord leng th measure ments; see [Scm93], [CB03] and [Ser 84], for example. Blaschk e asked whether the distribution of the lengths of chords (in all directions ) of a conv ex bo dy characterize s the b o dy , but Mallows and Cla rk [MC70] prov e d Date : No vem b er 9, 2021. 2000 Mathematics Subject Classiﬁc ation. Primar y 60D05; Secondary 52A22, 42B10, 52B10, 52A38. Key wor ds and phr ases. A utocorrelation, cov ariogram, cross co v ariogram, cut-and-pro ject sc heme, geometric tomograph y , phase retriev al, quasicrystal, set cov ariance. 1 2 GABRIELE BIANCHI that this is false even for co n vex p o lygons. In fact (see [Nag93]) the cov ariogr am problem is equiv alent to the problem of determining a con vex bo dy from all its separate chord le ng th distributions, one for ea ch directio n u ∈ S n − 1 . The cov a riogra m pro blem app ear s in other contexts. Adler and Pyke [AP91] asked in 199 1 whether the distribution of the diﬀerence X − Y of indep endent random v ar ia bles X and Y uniformly distr ibuted ov e r K determines K , up to translations and reﬂections . Since the conv olution in (1.1) is, up to a multiplicativ e factor, the pro bability density of X − Y , this problem is equiv alent to the cov ari- ogram pr oblem. Th e same a uthors [AP97] ﬁnd the cov ariogra m pr oblem relev ant also in the s tudy of scanning Brownian pro ces ses and of the equiv alence of measures induced by these pro cess es for diﬀeren t ba se se ts . The c ov ar iogra m problem is a s pe c ial case of the phase r et rieval pr oblem in F ourier analysis . This problem in volves determining a function f from the modulus of its F ourier transform b f . Since phase and amplitude are in general indep endent of each other, in order to so lve the phase retriev al pr oblem one must use additiona l in- formation constraining the admissible solutions f ; see [KST95] and [San85]. T a king F ourier transforms in (1.1) and using the relation d 1 − K = c 1 K , we obtain (1.2) c g K = c 1 K d 1 − K = | c 1 K | 2 . Thu s the pha s e retriev al pr oblem, restricted to the class of characteris tic functions of convex b o dies, reduces to the cov ariogra m problem. In X-ray c r ystallogr aphy the atomic structure of a cr ystal is to b e found from diﬀraction image s . As Rosenblatt [Ros84] explains , “Her e the phase retr iev al prob- lem arises be cause the mo dulus of a F ourier transform is all that can usually b e measured after diﬀraction o c c urs.” A conv enient way of describing many imp ortant examples of qua sicrystals is via the “cut a nd pro ject scheme”; s e e [BM04]. Here to the atomic str ucture, repres ent ed by a discr ete set S contained in a spa ce E , is as- so ciated a lattice N in a hig he r dimensio nal space E × E ′ and a “window” W ⊂ E ′ (whic h in many ca ses is a con vex set). Then S coincides with the pro jection on E o f the po ints of the lattice N which belong to W × E . In many examples the lattice N can be determined by the diﬀractio n image. T o determine S it is how ever nece ssary to know W : the cov a riogr am problem en ters at this p oint, since the cov ariogra m of W can be obtained b y the diﬀraction image; see [BG07]. In convex geometry the cov ariogra m app ears in s everal con tex ts . F or instance it has a central role in the pro of o f the Rogers- Shephard inequalit y [Sch93, Th. 7.3.1]. Moreov er the lev el sets of g K , which are conv ex and ar e called c onvolution b o dies , hav e b een studied [Tso 97] and are rela ted to the pro jection b o dy of K . A discr ete version o f the cov a riogra m problem has b een considere d [GGZ05]. In [GZ9 8] the cov ar iogra m problem w a s tra nsformed to a q uestion for the r adial me an b o dies . The ﬁrst contribution to Ma theron’s questio n was made b y Nagel [Na g93] in 1993, w ho co nﬁrmed Mather o n’s co njecture for all conv ex p olygons . Other partia l results tow ards the complete conﬁr mation of this co njecture in the pla ne hav e b een prov ed by Schmitt [Scm93 ], Bianchi, Segala and V o lˇ ciˇ c [BSV00], Bia nchi [Bia05] and Averko v a nd Bianchi [AB07]. In general, Matheron’s conjecture is false, as the author [Bia05] proved b y ﬁnding counterexamples in R n , for any n ≥ 4. Indeed, the cov ariogr am of the Ca r tesian pro duct of co nv ex sets K ⊂ R k and L ⊂ R m is the pro duct of the cov a riogr ams o f K a nd L . Thus K × L a nd K × ( − L ) have equal cov ar iogra ms. How ever, if neither K nor L is centrally s ymmetric, then K × L c a nnot be obtained from K × ( − L ) through a tr anslation or a reﬂection. T o satisfy these requirements the dimension of both sets must b e at least tw o and th us the dimensio n of the counterexamples THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 3 is at lea st four. W e note that these counterexamples can b e p olytop es but not C 1 bo dies. F or n -dimensiona l conv ex p olytop es P , Go o dey , Schneider and W eil [GSW97] prov e that if P is simplicial and P and − P are in general r elative p ositio n, the cov ar iogra m determines P . Up till now this was the only p os itive result av ailable in dimension n ≥ 3 (b eside the po sitive r esult for a ll centrally symmetric co nvex bo dies). In R 3 the “Car tesian pro duct” c o nstruction do es not apply b ecaus e any o ne- dimensional conv e x set is centrally sy mmetr ic. F or the cla s s of three-dimensional conv ex p o lytop es we are able to conﬁrm Ma theron’s conjecture. This a nswer, to- gether with the counterexamples for n ≥ 4, completely settles the cov ariogr am problem fo r conv ex p olytop es. Theorem 1 . 1. L et P ⊂ R 3 b e a c onvex p olytop e with non-empty interior. Then g P determines P , in the class of c onvex b o dies in R 3 , u p to tr anslations and r eﬂe ctions. Given a face F of a conv ex p olyto pe P ⊂ R 3 , co ne ( P , F ) denotes the supp ort cone to P a t F ( see the next section for all unexpla ined deﬁnitions). If w ∈ S 2 we denote by P w the unique prop er face o f P such that the rela tive interior of its normal cone contains w . A ma jor step in the pro of o f Theo rem 1 .1 is the following result. Theorem 1. 2. L et P and P ′ b e c onvex p olytop es in R 3 with non-empty interior such that g P = g P ′ . If w ∈ S 2 then, p ossibly after a tr anslation or a r eﬂe ction of P ′ that may dep end on w , P w = P ′ w ; (1.3) cone ( P, P w ) = cone ( P ′ , P ′ w ) . (1.4) Let P and Q b e conv ex p olyto pe s in R n , le t F b e a prop er face of P , a nd let G be a pro p er face of Q . W e say tha t F and G a r e isothetic if G is a translate of F and cone ( P, F ) = co ne ( Q, G ) . Given co nvex p olytop es P 1 , P 2 , Q 1 and Q 2 in R n we say that ( P 1 , P 2 ) and ( Q 1 , Q 2 ) are synisothetic if given a n y proper face F of P j , for some j = 1 , 2 , there is a pr o p e r face G of Q k , fo r so me k = 1 , 2 (and co nv erse ly ), s uch that F and G ar e isothetic. The term synisothetic was sugges ted by P . McMullen. The pr evious theor em can b e rephr ased in these terms: If g P = g P ′ , then ( P , − P ) and ( P ′ , − P ′ ) a re synisothetic. In or der to prove Theo rem 1.2 we inv estiga te tw o r elated pr oblems. The presence of parallel facets of P causes diﬃculties (eliminated by the sp ecial assumption in [GSW97] that P and − P a r e in g eneral relative p ositio n). T o deal with this, A. V ol ˇ ciˇ c and R. J. Gardner p osed a gener alization of Ma theron’s question we ca ll the cr oss c ovario gr am pr oblem . T o explain this, some ter minology is needed. Given t wo conv ex s ets K and L in R n , the cr oss c ovario gr am g K,L is the function g K,L ( x ) = λ n ( K ∩ ( L + x )) where x ∈ R n is such that λ n ( K ∩ ( L + x )) is ﬁnite. Let K , L , K ′ and L ′ be co n vex sets in R n . W e ca ll ( K, L ) and ( K ′ , L ′ ) t rivial asso ciates if one pair is obtained by the other one v ia a combination of the op e r - ations whic h leav e the cr oss cov ar iogram unch anged; s ee Section 2 for the pr ecise deﬁnition. Problem 1.3 (Cr oss cov ariog ram problem for p olyg ons) . D o es the cr oss c ovari- o gr am of the c onvex p olygons K and L determine the p air ( K , L ) , among al l p airs of c onvex b o dies, up to trivial asso ciates? 4 GABRIELE BIANCHI One co nnection b etw e e n cov ariogr a m and cro s s cov ariogra m lies in the obs er- v ation that if F and G denote parallel facets of a co nvex po lytop e P ⊂ R n , the “singular part” of some second order distributional deriv ative of g P provides b oth g F, G 0 and g F + g G 0 , where G 0 is the or thogonal pro jection of G o n the hype r plane which contains F . W e pr ov e tha t the infor mation g iven b y these t wo functions ca n be decoupled a nd pr ovides b oth g F and g G 0 , up to a n exchange b etw een them. When n = 3, in view of the conﬁrmation of Matheron’s conjecture in the pla ne, g P provides b oth F and G 0 , up to an exch ange b etw een them and up to tra nslations or reﬂections o f F and of G 0 . Ho wever, all this is not s uﬃcie nt for our purp ose and a detailed study o f Pro blem 1.3 is needed; see Rema rks 4.4 and 9.3 for further comments. The answer to Problem 1.3 is nega tive as Examples 3 .4 and 3.5 s how (see Fig- ures 1 a nd 2). F or ea ch choice of some real parameter s ther e exist four pairs o f parallelog rams ( K 1 , L 1 ) , . . . , ( K 4 , L 4 ) such that, for i = 1 , 3, g K i , L i = g K i +1 , L i +1 but ( K i , L i ) is not a trivial asso ciate of ( K i +1 , L i +1 ). P roblem 1.3 is co mpletely solved by Bianchi [Bia], which prov es that, up to an aﬃne trans formation, the previous counterexamples are the only ones. Theorem 1. 4 ([Bia]) . L et K , L b e c onvex p olygons and K ′ , L ′ b e planar c onvex b o dies with g K,L = g K ′ ,L ′ . Ass ume that ther e is no aﬃne tr ansformation T and no diﬀer ent indic es i, j , with either i, j ∈ { 1 , 2 } or i, j ∈ { 3 , 4 } , such that ( T K, T L ) and ( T K ′ , T L ′ ) ar e trivial asso ciates of ( K i , L i ) and ( K j , L j ) , r esp e ct ively. Then ( K, L ) is a trivial asso ciate of ( K ′ , L ′ ) . This result states that the information provided by the cross cov a riogra m of conv ex po lygons is so rich as to determine not only o ne unknown b o dy , as requir ed by Mather on’s co njecture, but tw o bo dies, with a few exceptions. The s econd problem is in some sense dual to the ﬁrst one and has b een intro duced by Mani-Levitsk a [Man01]. Let A and B b e conv ex p olyhedral cones in R 3 , with ap ex the orig in O and A ∩ B = { O } . Problem 1.5 (Cro ss cov ariogra m problem for cones) . Do es t he cr oss c ovario gr am of A and B determine the p air ( A, B ) , among al l p airs of c onvex c ones, up t o trivial asso ciates? Prop ositio n 5 .1 provides an answer to Pro blem 1.5, while Bianchi [Bia] (se e Lemma 3 .2 in this pap er) solves completely the corre sp o nding problem in the plane, describing also s ome situations o f non-unique determination. The techniques that we use to prov e Pro po sition 5 .1 rely on t wo main ingredients. The ﬁrst o ne is an analysis of the set where g A,B is not C 3 . The second ingredient is the observ ation that a suitable seco nd or der mixed deriv ative of g A,B provides ce r tain X -ray func- tions of the co nes. Some results regar ding the determination of conv ex p oly hedral cones fr om their X -ray functions are also needed (see [Bia 0 8]). The pro of of Theorem 1.1 also requires the study o f the structure of ∂ P ∩ ∂ P ′ , when P a nd P ′ are con vex p olytop es in R 3 and ( P, − P ) and ( P ′ , − P ′ ) a re s y niso- thetic. This study is cont ained in Sec tio n 7 while The o rem 1.1 is proved in Section 8. In Section 9 the counterexamples in dimension n ≥ 4 are pres ent ed in ter ms of decomp osition of a con vex b o dy in to dir ect summands, and the relation b etw ee n the cov ar iogra m a nd this deco mpo sition is studied. Theorem 9.1 classiﬁes the conv ex bo dies which hav e cov ariogr am equal to one o f the co unt erexamples . In view of all this the right fo rm to ask Ma theron’s pro blem for n -dimensional co nvex p olytop es P , when n ≥ 4, is with the restriction to dir e ctly inde c omp osable P . Ackno wledgements. W e are extremely grateful to G. Averk ov, R. J. Gardner and P . Gr onchi for reading lar ge pa rts o f this ma n uscript and suggesting many ar- guments that simpliﬁed and clariﬁed some of the pro ofs. W e also thank A. Zastrow, THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 5 for a v er y useful disc us sion regarding Lemma 7.1 0, and P . Mani-Levitsk a, for giving us his unpublished note [Man01]. 2. Definitions, not a tion s and preliminaries F or conv enience of the rea de r , we rep e a t here a ll the deﬁnitions already intro- duced. As usual, S n − 1 denotes the unit spher e in R n , centered at the origin O . F or x , y ∈ R n , k x k is the Euclidean norm of x and x · y denotes scalar pr o duct. F or δ > 0, B ( x, δ ) denotes the ope n ball in R n centered at x and with radius δ . If u ∈ S n − 1 , u ⊥ denotes the ( n − 1)-dimensio nal subspace orthogo nal to u . If A ⊂ R n we denote b y int A , cl A , ∂ A and co nv A the int erior , clo sur e , b oundary and c onvex hul l o f A , resp ectively . The char acteristic funct ion o f A is denoted by 1 A . The r eﬂection of A in the origin is − A . With the symbo l A | π we denote the orthogo nal pro jectio n of A on the a ﬃne space π . Mor eov er w e deﬁne p os A = { µx : x ∈ A, µ ≥ 0 } . The s ymbol λ n denotes n -dimensional Leb esgue meas ure, while H k denotes k -dimensional Haus dorﬀ meas ure in R n , w he r e 0 ≤ k ≤ n . Convex sets. A c onvex b o dy K ⊂ R n is a compact c o nv ex set with no n-empty int erior. The sym b ol a ﬀ K stands for the aﬃne hul l of K ; dim K is the dimensio n of aﬀ K . The sym b o ls re lbd K and relint K indicate respectively the r elative b oundary and the r elative interior of K . The diﬀer enc e b o dy of K is deﬁned by D K = K + ( − K ) = { x − y : x, y ∈ K } . The supp ort function o f K is deﬁned, for x ∈ R n , by h K ( x ) = sup { x · y : y ∈ K } . The Steiner p oint of K is deﬁned by s ( K ) = (1 /λ n ( B (0 , 1))) R S n − 1 h K ( u ) ud H n − 1 ( u ). W e write (2.1) K = K 1 ⊕ · · · ⊕ K s if K = K 1 + · · · + K s for suitable conv ex bo dies K i lying in linear subspaces E i of R n such that E 1 ⊕ · · · ⊕ E s = R n . If a representation K = L ⊕ M is only po ssible with dim L = 0 or dim M = 0 then K is dir e ctly inde c omp osable . E ach K , with dim K ≥ 1, has a representation as in (2 .1), with dim K i ≥ 1 and K i directly indecomp osable, whic h is unique up to the order of the summands. Given x, y ∈ R n , we write [ x, y ] for the segment with endp oints x and y . Given a conv ex b o dy K ⊂ R 2 and a, b ∈ ∂ K the symbol [ a, b ] ∂ K denotes { p ∈ ∂ K : a ≤ p ≤ b } in counterclo ckwise order on ∂ K and ( a, b ) ∂ K denotes the co rresp onding op en arc. W e will r efer to a as the lower endp oint of the a rc a nd to b a s its upp er en dp oint . The X-r ay o f a conv ex set K ⊂ R n with resp ect to u ∈ S n − 1 is the function which asso cia tes to ea ch line l para llel to u the le ngth of K ∩ l . The − 1 - chor d function of K a t p ∈ R n \ cl K is deﬁned, for each line l through p , by Z K ∩ l k x − p k − 2 dλ 1 ( x ) . Polytop es. Let P b e a conv e x p olyto p e in R n . As us ual the 0- , 1- and ( n − 1 )- dimensional faces ar e called vertices, edges a nd facets, resp ectively . Given a face F of P the normal c one of P at F is denoted by N ( P , F ) and is the se t of all outer normal vectors to P at x , where x ∈ relint F , to gether with O . The supp ort c one of P at F is the set cone ( P, F ) = { µ ( y − x ) : y ∈ P , µ ≥ 0 } , where x ∈ relint F . Neither deﬁnitions depe nd on the choice of x . Two faces F and G o f P are antip o dal if relint N ( P, F ) ∩ ( − relint N ( P, G )) 6 = ∅ . Given u ∈ S n − 1 the exp ose d fac e of P in dir e ction u is P u = { x ∈ P : x · u = h P ( u ) } . 6 GABRIELE BIANCHI It is the unique prop er face of P such that the relative interior of its nor mal cone contains u . W e w ill rep eatedly use the following iden tities, proved in [Sch93, Th. 1 .7.5(c)] a nd v alid for all u ∈ S n − 1 and all conv ex p o ly top es P , P ′ in R n : (2.2) ( P + P ′ ) u = P u + P ′ u ; ( D P ) u = P u + ( − P ) u . Given a face G of P , the meaning of Σ + ( G ), x + ( G ), x − ( G ), P G and o f p o sitive, negative or neutral face is intro duce d in Deﬁnitions 7.4 and 7.5. See the s tatement of Lemma 7.1 3 for the meaning of Σ − ( G ). W e say that P a nd − P are in gener al r elative p osition if dim P w ∩ ( P − w + x ) = 0 for each w ∈ S n − 1 and for each x ∈ R n . In this pap er the term c one always means cone with apex O . A poly hedral conv ex cone is dihe dr al if it is the intersection of tw o closed half-spaces. If F is an edge of a three-dimensional con vex p o lytop e P , then co ne ( P , F ) is dihedr al. A conv ex c o ne is p ointe d if its a p e x is a vertex. Synisothesis. Let P and Q b e conv ex p olytop es in R n , let F b e a prope r face of P , and let G b e a pr op er face o f Q . W e say that F and G are isothetic if G is a translate of F and cone ( P, F ) = co ne ( Q, G ) . Given co nvex p olytop es P 1 , P 2 , Q 1 and Q 2 in R n we say that ( P 1 , P 2 ) and ( Q 1 , Q 2 ) are synisothetic if given an y prop er face F of P 1 or of P 2 there is a prop er face G of Q 1 or of Q 2 (and con versely) such that F and G are isothetic. Covario gr am and trivial asso ciates. Let K , L , K ′ and L ′ be conv e x sets in R n . The cr oss c ovario gr am g K,L is the function g K,L ( x ) = λ n ( K ∩ ( L + x )), where x ∈ R n is such that λ n ( K ∩ ( L + x )) is ﬁnite. It is evident that g K,K = g K and that, fo r a ny x ∈ R n , g K,L = g K + x ,L + x = g − L, − K . W e call ( K , L ) and ( K ′ , L ′ ) t rivial asso ciates if either ( K, L ) = ( K ′ + x, L ′ + x ) or ( K, L ) = ( − L ′ + x, − K ′ + x ), for so me x ∈ R n . It is easy to prov e that supp g K = D K , supp g K,L = K + ( − L ) , (2.3) g K = 1 K ∗ 1 − K , g K,L = 1 K ∗ 1 − L , (2.4) c g K = | c 1 K | 2 and d g K,L = c 1 K · c 1 L , (2.5) where s upp f and b f denote resp ectively the suppo rt a nd the F o urier tra nsform o f the function f . 3. The cross cov ariogram pr oblem f or pol ygons This sectio n r ecalls so me results pr oved in [Bia] and needed in this pap er. Let ( ρ, θ ) denote p olar co or dinates. F or br e vity , given α, β ∈ [0 , 2 π ] with α < β , we write { α ≤ θ ≤ β } for the cone { ( ρ, θ ) : α ≤ θ ≤ β } . Example 3.1. Let A 1 = { 0 ≤ θ ≤ 3 π / 4 } , B 1 = −{ π / 4 ≤ θ ≤ π / 2 } , A 2 = { 0 ≤ θ ≤ π / 4 } and B 2 = −{ π / 2 ≤ θ ≤ 3 π / 4 } . W e hav e {A 1 , −B 1 } 6 = {A 2 , −B 2 } and g T A 1 , T B 1 = g T A 2 , T B 2 , fo r a ny non-s ingular aﬃne transformatio n T . Lemma 3. 2. L et A , B , A ′ and B ′ b e p ointe d close d c onvex c ones in R 2 with non- empty int erior and ap ex t he origin O such that int A ∩ int B = ∅ . The identity g A,B = g A ′ ,B ′ holds if and only if one of t he fol lowing c ases o c curs: (i) { A, − B } = { A ′ , − B ′ } ; (ii) ther e ex ist a line ar tr ansformation T and i, j ∈ { 1 , 2 } , i 6 = j , such that (3.1) {T A, − T B } = {A i , −B i } and {T A ′ , −T B ′ } = {A j , −B j } . Remark 3. 3. O bserve that int A 1 ∩ int ( −B 1 ) 6 = ∅ and in t A 2 ∩ int ( −B 2 ) = ∅ . Thus if in t A ∩ int ( − B ) and int A ′ ∩ int ( − B ′ ) are b oth empty or both no n-empty , then Lemma 3.2 implies { A, − B } = { A ′ , − B ′ } . THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 7 K 2 L 2 K 1 L 1 Figure 1. Up to aﬃne transforma tions, ( K 1 , L 1 ) a nd ( K 2 , L 2 ) are the only pairs of conv ex p olygons with equal cross cov a riogr am which ar e not synisothetic. K 3 L 3 K 4 L 4 Figure 2. Up to aﬃne transforma tions, ( K 3 , L 3 ) a nd ( K 4 , L 4 ) are the only pairs of conv ex p olygons with equal cross cov a riogr am which ar e syniso thetic and are no t tr ivial a sso ciates. Example 3. 4. Let α, β , γ and δ b e p ositive real n um b ers, let y ∈ R 2 and let I 1 = [( − 1 , 0) , (1 , 0 )], I 2 = (1 / √ 2) [( − 1 , − 1) , (1 , 1)], I 3 = [(0 , − 1) , (0 , 1 )] a nd I 4 = (1 / √ 2) [(1 , − 1 ) , ( − 1 , 1 )]. W e deﬁne four para llelogra ms as follows: K 1 = αI 1 + β I 2 ; L 1 = γ I 3 + δ I 4 + y ; K 2 = αI 1 + δ I 4 and L 2 = β I 2 + γ I 3 + y . See Fig. 1. The pairs ( K 1 , −L 1 ), ( K 2 , −L 2 ) are not synisothetic (no vertex o f a po lygon in the second pa ir has a supp or t cone equal to the supp or t cone of the top vertex o f L 1 or to its reﬂection). Mor eov er g K 1 , L 1 = g K 2 , L 2 . Example 3.5. Let α, β , γ and δ b e p os itive real num b ers, let m ∈ R , y ∈ R 2 and let I ( m ) = (1 / √ 1 + m 2 ) [( − m, − 1 ) , ( m, 1)]. Assume either m = 0, α 6 = γ a nd β 6 = δ or else m 6 = 0 and α 6 = γ . W e deﬁne four paralle lo grams as f ollows: K 3 = αI 1 + β I 3 ; L 3 = γ I 1 + δ I ( m ) + y ; K 4 = γ I 1 + β I 3 and L 4 = αI 1 + δ I ( m ) + y . See Fig. 2. W e have g K 3 , L 3 = g K 4 , L 4 and the pairs ( K 3 , −L 3 ) and ( K 4 , −L 4 ) ar e synisothetic. How ever, ( K 3 , L 3 ) a nd ( K 4 , L 4 ) a re no t tr ivial a sso ciates. Lemma 3. 6. L et A , B , C and D b e c onvex c ones in R n with ap ex the origin O . Assume that e ach o f them eithe r c oincides with { O } or has non-empty interior, and, mor e over, A ∪ B ⊂ { ( x 1 , x 2 , . . . , x n ) : x n ≥ 0 } , A ∩ { x n = 0 } = B ∩ { x n = 0 } = { O } , C ∪ D ⊂ { x n ≤ 0 } and conv ( C ∪ D ) is p ointe d. If g A,C + g B ,D = g A,D + g B ,C then either A = B or C = D . 4. Determining the f aces o f P : pr oof of (1.3) in Theorem 1.2 The result reg a rding g F 0 + g G 0 in next prop o sition was ﬁrst obs erved by K . Ru- ﬁbach [Ruf01 , p.14]. Prop ositi o n 4.1 . L et P ⊂ R n b e a c onvex p olytop e with non-empty int erior, let w ∈ S n − 1 , F = P w and G = P − w . The c ovario gr am g P determines b oth g F 0 + g G 0 and g F 0 ,G 0 , wher e F 0 = F | w ⊥ and G 0 = G | w ⊥ . 8 GABRIELE BIANCHI The p ossibility of proving P rop osition 4 .1 using the ex pression of the sec o nd order distributional deriv ative of g P computed in next lemma was sug gested by G. Averko v. Let C ∞ 0 ( R n ) deno te the cla ss o f inﬁnitely diﬀerentiable functions on R n with co mpact supp or t. Lemma 4 .2. L et P ⊂ R n b e a c onvex p olytop e with non-empty interior. L et F 1 , . . . , F m b e its fac ets, ν i b e t he unit outer normal of P at F i , for i = 1 , . . . , m , let w ∈ S n − 1 and let I p = { ( i, j ) : F i is p ar al lel to F j } and I np = { ( i, j ) : F i is not p ar al lel to F j } . Then, for φ ∈ C ∞ 0 ( R n ) , we have (4.1) − ∂ 2 g P ∂ w 2 ( φ ) = X ( i,j ) ∈ I np w · ν i w · ν j p 1 − ( ν i · ν j ) 2 Z R n H n − 2 ( F i ∩ ( F j + z )) φ ( z ) dz + X ( i,j ) ∈ I p w · ν i w · ν j Z F i − F j H n − 1 ( F i ∩ ( F j + z )) φ ( z ) d H n − 1 ( z ) . Both sum s in the right hand side of (4.1) ar e uniquely determine d by g P . Pr o of. Let δ F i ( φ ) = R F i φ ( x ) d H n − 1 ( x ). It is easy to prov e tha t ( ∂ 1 P /∂ w )( φ ) = − P m i =1 w · ν i δ F i ( φ ). F or instance, [Ho r 83, p.60] proves the cor resp onding formula for sets with C 1 bo undary and the f ormula for P ca n b e pro ved b y an approximation argument. If ( P n ) is a sequence of conv ex b o dies with C 1 bo undary con verging to P in the Hausdo rﬀ metric, then ( ∂ 1 P /∂ w )( φ ) = lim n ( ∂ 1 P n /∂ w )( φ ), by dominated conv er g ence Theorem [EV92, p. 2 0]. Thus, if ν P n denotes the o uter norma l to ∂ P n , we have ∂ 1 P ∂ w ( φ ) = lim n ∂ 1 P n ∂ w ( φ ) = − lim n Z ∂ P n w · ν P n ( x ) φ ( x ) d H n − 1 ( x ) = − m X i =1 w · ν i δ F i ( φ ) . Since ∂ 1 P /∂ w has co mpact suppor t and g P = 1 P ∗ 1 − P we ca n write (4.2) ∂ 2 g P ∂ w 2 ( φ ) =  ∂ 1 P ∂ w ∗ ∂ 1 − P ∂ w  ( φ ) = − m X i,j =1 w · ν i w · ν j ( δ F i ∗ δ − F j )( φ ) . Assume that F i and F j are parallel and choose a Car tesian co ordinates system so that F i ⊂ { x ∈ R n : x 2 = 0 } and F j ⊂ { x : x 2 = α } , where x = ( x 1 , x 2 ) ∈ R n − 1 × R and α ∈ R . W e have ( δ F i ∗ δ − F j )( φ ) = Z F i Z − F j φ ( x + y ) d H n − 1 ( y ) ! d H n − 1 ( x ) = Z R n − 1 1 F i ( x 1 , 0)  Z R n − 1 1 − F j ( y 1 , − α ) φ ( x 1 + y 1 , − α ) dy 1  dx 1 = Z R n − 1  Z R n − 1 1 F i ( x 1 , 0)1 − F j ( z 1 − x 1 , − α ) dx 1  φ ( z 1 , − α ) dz 1 . THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 9 Since 1 F i ( x 1 , 0)1 − F j ( z 1 − x 1 , − α ) = 1 F i ∩ ( F j +( z 1 , − α )) ( x 1 , 0) and F i − F j ⊂ { x : x 2 = − α } , we have (4.3) ( δ F i ∗ δ − F j )( φ ) = Z R n − 2 H n − 1 ( F i ∩ ( F j + ( z 1 , − α ))) φ ( z 1 , − α ) dz 1 = Z F i − F j H n − 1 ( F i ∩ ( F j + z )) φ ( z ) d H n − 1 ( z ) . Assume n ≥ 3, that F i and F j are not para llel and cho ose a Cartesian co or dinates system so that F i ⊂ { x ∈ R n : x 3 = 0 } a nd F j ⊂ { x : x 1 = αx 3 } , wher e x = ( x 1 , x 2 , x 3 ) ∈ R × R n − 2 × R and α ∈ R . W e have ( δ F i ∗ δ − F j )( φ ) = p 1 + α 2 Z R × R n − 2 1 F i ( x 1 , x 2 , 0)  Z R n − 2 × R 1 − F j ( αy 3 , y 2 , y 3 ) φ ( x 1 + αy 3 , x 2 + y 2 , y 3 ) dy 2 dy 3  dx 1 dx 2 = p 1 + α 2 Z R n  Z R n − 2 1 F i ( z 1 − αz 3 , x 2 , 0) 1 − F j ( αz 3 , z 2 − x 2 , z 3 ) dx 2  φ ( z ) dz , where z denotes ( z 1 , z 2 , z 3 ) a nd in the last in teg r al we hav e used the c hange of v ariable ( x 1 , y 2 , y 3 ) = ( z 1 + αz 3 , z 2 − x 2 , z 3 ). Since 1 F i ( z 1 − αz 3 , x 2 , 0)1 − F j ( αz 3 , z 2 − x 2 , z 3 ) = 1 F i ∩ ( F j + z ) ( z 1 − αz 3 , x 2 , 0) and F i ∩ ( F j + z ) ⊂ { x : x 1 = z 1 − αz 3 , x 3 = 0 } , the inner integral in the last line of the previous form ula equals H n − 2 ( F i ∩ ( F j + z )). Thu s, s inc e √ 1 + α 2 = (1 − ( ν i · ν j ) 2 ) − 1 / 2 , we have (4.4) ( δ F i ∗ δ − F j )( φ ) = 1 q 1 − ( ν i · ν j ) 2 Z R n H n − 2 ( F i ∩ ( F j + z )) φ ( z ) dz . When n = 2 formula (4.4) is prov ed as ab ov e by adapting the notations. The formulas (4.2 ), (4 .3) and (4 .4) imply (4.1). By (4.1), ther e exists C ∈ R such that − ( ∂ 2 g P /∂ w 2 , φ ) ≤ C sup R n | φ | for each φ ∈ C ∞ 0 ( R n ). Therefore , by [Hor8 3, Th. 2 .1.6], the distribution − ∂ 2 g P /∂ w 2 has an unique extension to a bo unded linear functional on C c ( R n ), the spac e of func- tions o n R n which v anish at inﬁnity endow ed w ith the supr emum norm. The Riesz representation Theorem [E V92, p. 49] implies the ex istence of a Radon measure µ and a µ -measurable function σ , with | σ | = 1 µ - almost everywhere, suc h that − ( ∂ 2 g P /∂ w 2 , ψ ) = R R n ψ σ dµ for each ψ ∈ C c ( R n ). By Leb esgue decomp osition Theorem [EV92, p. 42] the measure µ has an unique decomp osition µ = µ ac + µ s , where µ ac is abso lutely co ntin uous with r esp ect to λ n and µ s and λ n are mutually singular. The ﬁrst sum in the right hand side of (4.1) coincides with R φ σ dµ ac , while the seco nd sum coincides with R φ σ dµ s . Both sums are thus uniquely deter- mined by g P .  Pr o of of Pr op osition 4.1. Le t F i , ν i and I p be as in the statement of Lemma 4.2. Consider the distribution deﬁned by the s econd sum in (4.1). This distribution determines its supp ort S ( P, w ) = ∪ ( i,j ) ∈ I p : ν i · w 6 =0 ( F i − F j ) a nd (4.5) X ( i,j ) ∈ I p w · ν i w · ν j H n − 1 ( F i ∩ ( F j + x )) , for each x ∈ S ( P, w ). Clearly we have S ( P , w ) ⊂ D P . If F i and F j are parallel and i 6 = j then ν i = − ν j and, by (2.2), F i − F j is the facet ( D P ) ν i of D P . Moreover F i − F i ⊂ ν ⊥ i . Thus we hav e (4.6) S ( P, w ) ∩ int D P = ∪ i : ν i · w 6 =0 ( F i − F i ) ⊂ ∪ i : ν i · w 6 =0 ν ⊥ i 10 GABRIELE BIANCHI and H n − 1 ( S ( P, w ) ∩ ν ⊥ i ) > 0 when ν i · w 6 = 0. If H n − 1 ( S ( P, w ) ∩ w ⊥ ) = 0 then neither F nor G are fac ets o f P and g F 0 + g G 0 ≡ 0. Now assume H n − 1 ( S ( P, w ) ∩ w ⊥ ) > 0 . In this ca se w coincides, up to the sign, with o ne of the ν i . If x ∈ ( w ⊥ \ ∪ ν j 6 = ± w ν ⊥ j ) ∩ int D P then, by (4 .6), the expr e ssion in (4.5) coincides with H n − 1 ( F ∩ ( F + x )) + H n − 1 ( G ∩ ( G + x )) = H n − 1 ( F 0 ∩ ( F 0 + x )) + H n − 1 ( G 0 ∩ ( G 0 + x )) = g F 0 ( x ) + g G 0 ( x ). On the other hand, if x ∈ w ⊥ \ D P we ha ve g F 0 ( x ) + g G 0 ( x ) = 0, b eca use P ∩ ( P + x ) = ∅ implies F ∩ ( F + x ) = ∅ and G ∩ ( G + x ) = ∅ . Since g F 0 + g G 0 is cont inuous, this function is determined for all x ∈ w ⊥ by contin uity . Consider ( D P ) w . If it is not cont ained in S ( P , w ) then either F or G is not a facet of P and g F 0 ,G 0 ≡ 0 . If it is c ontained then F = F i , G = F j , for s ome i and j with i 6 = j , a nd ( D P ) w = F i − F j . In this case the express ion in (4.5) coincides with H n − 1 ( F i ∩ ( F j + x )). Knowing this function for each x ∈ ( D P ) w is equiv a lent to knowing g F 0 ,G 0 .  Lemma 4.3. L et F , F ′ , G and G ′ b e c onvex b o dies in R n with int F 6 = ∅ . If g F = αg F ′ , for some α 6 = 0 , then α = 1 . If (4.7) ( g F + g G = g F ′ + g G ′ , g F, G = g F ′ ,G ′ then either g F = g F ′ and g G = g G ′ , or else g F = g G ′ and g G = g F ′ . Pr o of. Obse r ve that if K ⊂ R n is a conv ex b o dy then λ n ( K ) = g K (0) and λ 2 n ( K ) = Z R n Z R n 1 K ( y )1 K ( y − x ) dy dx = Z R n g K ( x ) dx (see (2.5) and [Sch93, p. 411 ]). Th us the identit y g F = αg F ′ implies λ n ( F ) = αλ n ( F ′ ), λ 2 n ( F ) = αλ 2 n ( F ′ ) a nd, as a co ns equence, α = 1. Let us prove the second cla im. Applying the F ourier tra nsform to the equalities in (4 .7) we arr ive, with the help of (2.5), to the system ( k c 1 F k 2 + k c 1 G k 2 = k c 1 F ′ k 2 + k c 1 G ′ k 2 k c 1 F k 2 k c 1 G k 2 = k c 1 F ′ k 2 k c 1 G ′ k 2 . Let ξ ∈ R n denotes the tr ansform v ar iable. F o r each ξ ∈ R n , the previous system implies that either we have k c 1 F ( ξ ) k = k c 1 F ′ ( ξ ) k and k c 1 G ( ξ ) k = k c 1 G ′ ( ξ ) k or else we have k c 1 F ( ξ ) k = k c 1 G ′ ( ξ ) k and k c 1 G ( ξ ) k = k c 1 F ′ ( ξ ) k . The alternative a prio ri may depe nd on ξ . The F o urier transfor m of a function with compa ct supp or t is analytic (see [Hor83, Th. 7.1 .14]) and therefore the squar ed mo duli of the pr evious transforms are analytic. Since any analytic function is determined by its v a lue s on a se t with a limit p oint, we conclude that the previo us alternative do es no t dep end on ξ . Going back to c ov ariog rams via F ourier inv ersion, this means that either g F = g F ′ and g G = g G ′ , o r else g F = g G ′ and g G = g F ′ .  Remark 4.4 . Let P , F , G , F 0 and G 0 be as in Pro po sition 4 .1. Assume n = 3 and F and G facets, let P ′ be a convex p olytop e with g P = g P ′ and let F ′ = P ′ w , G ′ = P ′ − w , F 0 ′ = F ′ | w ⊥ and G 0 ′ = G ′ | w ⊥ . Pro p o sition 4 .1, Lemma 4.3 and the po sitive answer to the cov ariogra m problem in the plane imply that, p ossibly after a reﬂection o f P ′ , F 0 ′ and G 0 ′ are trans lations or reﬂections r e sp ectively of F 0 and G 0 . Ruling out the p o s sibility that, say , F 0 = − F 0 ′ 6 = F 0 ′ and G 0 = − G 0 ′ 6 = G 0 ′ is a ma jor diﬃculty in the pro of of Theor em 1.2, and to ov ercome it we need Theorem 1.4. This p oss ibilit y cannot b e ov e r come when n ≥ 4; see Remark 9.3. THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 11 Pr o of of (1.3) in The or em 1.2. Let F = P w , G = P − w , F ′ = P ′ w and G ′ = P ′ − w . The relations (2.3) and (2.2) imply (4.8) F − G = ( D P ) w = ( D P ′ ) w = F ′ − G ′ . Up to a transla tion of P and P ′ , a r eﬂection of P ′ and an aﬃne tra nsformation, we may a ssume w = (0 , 0 , − 1), dim F ≥ dim G, dim F ′ ≥ dim G ′ , (4.9) F, F ′ ⊂ { x : x 3 = 0 } , G, G ′ ⊂ { x : x 3 = 1 } , (4.10) s ( F ) = s ( F ′ ) = O and s ( G ) = s ( G ′ ) = (0 , 0 , 1). (4.11) Here x = ( x 1 , x 2 , x 3 ) ∈ R 3 and we hav e used (4 .8 ) and the Minkowski-additivit y of the Steiner p oint (see [Sch93, p. 42]) to obta in (4.11). Let ε > 0 and let p ε = (0 , 0 , − 1 + ε ). W e hav e P ∩ ( P + p ε ) ⊂ { x : 0 ≤ x 3 ≤ ε } . W e study the asymptotic be haviour o f the volume of this set as ε tends to 0 + . Let G 0 = G | { x : x 3 = 0 } and G 0 ′ = G ′ | { x : x 3 = 0 } . Observe that O ∈ relint F ∩ relint G 0 and O ∈ relin t F ′ ∩ relint G 0 ′ , since s ( F ) = s ( F ′ ) = s ( G 0 ) = s ( G 0 ′ ) = O and the Steiner point of a conv ex bo dy b elo ngs to its r e la tive interior; see [Sch93, p. 43]. According to the dimensio n of F and G , w e distinguish the following cases ( c deno tes a po sitive c onstant which may v ar y from formula to formula). Case 1: F and G ar e fac ets. W e have g P ( p ε ) = ε λ 2 ( F ∩ G 0 ) + o ( ε ). Indeed P ∩ ( P + p ε ) c oincides (up to p olytop es of volume o ( ε 2 )) with the sum of the p oly gon F ∩ G 0 and the seg ment [ O , (0 , 0 , ε )]. Case 2: F is a fac et and G is an e dge. W e hav e g P ( p ε ) = c ε 2 λ 1 ( F ∩ G 0 ) + o ( ε 2 ). Indeed P ∩ ( P + p ε ) co incides (up to p oly to pe s o f volume o ( ε 3 )) with the sum of the s egment F ∩ G 0 and a triangle with edg e lengths prop or tional to ε cont ained in a plane orthogonal to F ∩ G 0 . Case 3: F is a fac et and G is a vertex . W e have g P ( p ε ) = c ε 3 , since P ∩ ( P + p ε ) is a pyramid with edg e lengths prop o rtional to ε . Case 4: F and G ar e p ar al lel e dges. W e hav e g P ( p ε ) = c ε 2 + o ( ε 2 ), b ecause P ∩ ( P + p ε ) coincides (up to p olytop es o f volume o ( ε 3 )) with the s um of the segment F ∩ G 0 and a quadrilatera l with edge lengths pr op ortional to ε , contained in a plane orthogonal to F ∩ G 0 . Case 5: F and G ar e non-p ar al lel e dges. W e hav e g P ( p ε ) = c ε 3 , b ecaus e P ∩ ( P + p ε ) is a tetr a hedron with edge lengths prop ortiona l to ε . Case 6: F is an e dge and G is a vertex. W e hav e g P ( p ε ) = c ε 3 , b ecause P ∩ ( P + p ε ) is a po lytop e with edge lengths pr op ortional to ε . Case 7: F and G ar e vertic es . This is the only case where ( D P ) w is a p oint. In Case 3 we have g F + g G 0 = g F 6≡ 0 while, in Ca s e 5, we have g F + g G 0 ≡ 0. case F , G main term of g P ( p ε ) ( D P ) w g F + g G 0 1 facet, facet λ 2 ( F ∩ G 0 ) ε facet 2 facet, edge c λ 1 ( F ∩ G 0 ) ε 2 facet 3 facet, v ertex c ε 3 facet 6≡ 0 4 parallel edges c ε 2 edge 5 non-para lle l edges c ε 3 facet ≡ 0 6 edge, vertex c ε 3 edge 7 vertex, vertex vertex The information summarised in the last thr ee columns of this table is provided by g P (recall Pr op osition 4.1 and (2 .3)). This informa tio n distinguishes each cas e from the others. T o conclude the pro of it suﬃces to show that in each case w e have F = F ′ and G = G ′ , p o ssibly a fter a reﬂection of P ′ ab out (0 , 0 , 1 / 2). 12 GABRIELE BIANCHI Case 1. Prop os ition 4.1 implies g F, G 0 = g F ′ ,G 0 ′ . If ( F, G 0 ) a nd ( F ′ , G 0 ′ ) a re trivial asso cia tes, then, p ossibly after a r e ﬂection of P ′ ab out (0 , 0 , 1 / 2), we hav e F = F ′ + y and G 0 = G 0 ′ + y , fo r some y ∈ { x : x 3 = 0 } . The as sumption (4.11) implies y = 0, becaus e s ( F ′ + y ) = s ( F ′ ) + y ; s ee [Sch93, p. 43]. Now assume that ( F, G 0 ) a nd ( F ′ , G 0 ′ ) a re not trivial a sso ciates. Theorem 1.4 states that ( F , G 0 ) and ( F ′ , G 0 ′ ) are r e sp ectively trivial a s so ciates o f ( T K i , T L i ) and ( T K j , T L j ), for some aﬃne transfo rmation T and diﬀerent indices i, j , with either i, j ∈ { 1 , 2 } o r i, j ∈ { 3 , 4 } . Pr op osition 4.1 implies g T K i + g T L i = g T K j + g T L j . Lemma 4 .3 and the po sitive answer to the cov ariog ram problem in the plane [AB] imply that e ither K i is a tra nslation o r a reﬂection of K j and L i is a translation or a reﬂection o f L j , or else K i is a translation or a r eﬂection o f L j and L i is a transla tion or a reﬂec tion of K j . This is cle a rly false. Case 2. W e hav e g F = g F + g G 0 = g F ′ + g G 0 ′ = g F ′ , which implies either F = F ′ or F = − F ′ (recall, ag ain, (4.11)). When F = F ′ , (4.8) implies G = G ′ , since the Minko wski a ddition satisﬁes a cancellation law. When F = − F ′ , (4 .8) implies (4.12) F ′ + G 0 = − F ′ + G 0 ′ . W e claim tha t G 0 = G 0 ′ (and F ′ = − F ′ ). Obser ve that (4.11) implies that O is the midp oint o f G 0 and of G 0 ′ . Identit y (4.12) implies, for each u ∈ S 1 , h F ′ ( u ) − h − F ′ ( u ) = h G 0 ′ ( u ) − h G 0 ( u ) . The function in the r ight hand s ide is even, since G 0 and G 0 ′ are o -symmetric. The function in the left hand s ide is o dd, since h − F ′ ( u ) = h F ′ ( − u ). Thus b oth fun ctions v anish a nd G 0 = G 0 ′ . Again, (4.12) implies F = F ′ . Cases 3, 6 a nd 7. In each of these ca ses G and G ′ are vertices and, by (4 .11), G = G ′ = { (0 , 0 , 1) } . Identit y (4.8) implies F = F ′ to o. Case 4. The face ( D P ) w determines the direction of the edges F and G and the sum of their le ngths, b ecause λ 1 (( D P ) w ) = λ 1 ( F ) + λ 1 ( G ). Thus F , G , F ′ and G ′ are parallel and λ 1 ( F ) + λ 1 ( G ) = λ 1 ( F ′ ) + λ 1 ( G ′ ). O n the other hand, if q ∈ { x : x 3 = 0 } is parallel to ( D P ) w we ha ve g P ( p ε + q ) = c ε 2 λ 1 ( F ∩ ( G 0 + q )) + o ( ε 2 ), where the strictly p o sitive constant c do es no t dep end on q (it dep ends o nly on the “op enings” o f the dihedral cones cone ( P , F ) a nd c o ne ( P, G )). Thus we have c λ 1 ( F ∩ ( G 0 + q )) = c ′ λ 1 ( F ′ ∩ ( G 0 ′ + q )) , where the constant c ′ may a priori diﬀer from c . The term c λ 1 ( F ∩ ( G 0 + q )) coincides with c λ 1 ( F ∩ G 0 ) when q satisﬁes 2 k q k ≤ α := max( λ 1 ( F ) , λ 1 ( G )) − min( λ 1 ( F ) , λ 1 ( G )), and it is strictly less than c λ 1 ( F ∩ G 0 ) when 2 k q k > α . Similar consideratio ns hold a ls o for c ′ λ 1 ( F ′ ∩ G 0 ′ ). Thus we hav e α = max( λ 1 ( F ′ ) , λ 1 ( G ′ )) − min( λ 1 ( F ′ ) , λ 1 ( G ′ )) . Thu s we hav e F = F ′ and G = G ′ , up to a r eﬂection o f P ′ ab out (0 , 0 , 1 / 2). Case 5. The face ( D P ) w is a para llelogra m and therefo r e has an unique de- comp osition as Minkowski sum o f tw o summands, except for the o r der of the sum- mands. Ther e fore (4.8) implies F = F ′ and G = G ′ , up to a reﬂection of P ′ ab out (0 , 0 , 1 / 2).  5. The cross cov ariogram pr oblem for co n es Let A , A ′ , B and B ′ be conv ex p olyhedr al cones in R 3 with no n-empty interior and such that (5.1) A, A ′ , − B , − B ′ ⊂ { x : x 3 ≥ 0 } , A ∩ { x : x 3 = 0 } = B ∩ { x : x 3 = 0 } = { O } , A ′ ∩ { x : x 3 = 0 } = B ′ ∩ { x : x 3 = 0 } = { O } , THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 13 where x = ( x 1 , x 2 , x 3 ) ∈ R 3 . Consider the poly gons (5.2) F = A ∩ { x : x 3 = 1 } , F ′ = A ′ ∩ { x : x 3 = 1 } , G = ( − B ) ∩ { x : x 3 = 1 } , G ′ = ( − B ′ ) ∩ { x : x 3 = 1 } . It is easy to see that in this setting A − B = c o nv ( A ∪ ( − B )) and A ′ − B ′ = conv ( A ′ ∪ ( − B ′ )). Ther efore, g A,B = g A ′ ,B ′ implies, b y (2.3), (5.3) H := conv ( F ∪ G ) = conv ( F ′ ∪ G ′ ) . Prop ositi o n 5.1 . L et A , A ′ , B , B ′ , F , F ′ , G , G ′ and H b e as ab ove, and assume g A,B = g A ′ ,B ′ . L et z 1 , . . . , z n denote the vertic es of H in c ount er clo ckwise or der. Assume that the fol lowing assumptions hold, for e ach i = 1 , . . . , n : (i) the p oint z i ∈ F ∩ G if and only if z i ∈ F ′ ∩ G ′ ; (ii) the se gment [ z i , z i +1 ] is an e dge of F or of G if and only if it is an e dge of F ′ or of G ′ ; (iii) if z i / ∈ F ∩ G , then the p olygon, b etwe en F and G , which c ontains z i c oincides in a neighb ourho o d of z i with t he p olygon, b etwe en F ′ and G ′ , which c ontains z i . Then either A = A ′ and B = B ′ or else A = − B ′ and B = − A ′ . T o prov e this result w e need some preliminary lemmas. 5.1. Cov ariogram, X -ra ys of cones and − 1 -chord functions of thei r s ec- tions. The next lemma is the cr ucia l result that co nnects cov ar iogram a nd X -rays. Lemma 5 . 2. Le t L ⊂ R 3 b e a dihe dr al c one, R 0 b e its e dge and R 1 and R 2 b e its fac ets. F or i = 1 , 2 , let v i ∈ S 2 ∩ relint R i . If ˜ A ⊂ R 3 is a c onvex c one, ˜ A ∩ L = { O } , and t , s ∈ R ar e chosen so that A ∩ ( L + tv 1 + sv 2 ) = ∅ or int A ∩ ( L + tv 1 + sv 2 ) 6 = ∅ , then (5.4) ∂ 2 ∂ s∂ t λ 3 ( ˜ A ∩ ( L + tv 1 + sv 2 )) = α λ 1 ( ˜ A ∩ ( R 0 + tv 1 + sv 2 )) . Her e α is a p ositive c onstant which do es not dep end on ˜ A . Pr o of. Assume L = { x : x 1 ≥ 0 , x 2 ≥ 0 } , v 1 = (1 , 0 , 0) and v 2 = (0 , 1 , 0). Standard calculus a rguments prove the formulas ∂ ∂ t λ 3 ( ˜ A ∩ { x : x 1 ≥ t, x 2 ≥ s } ) = − λ 2 ( ˜ A ∩ { x : x 1 = t, x 2 ≥ s } ) , ∂ ∂ s λ 2 ( ˜ A ∩ { x : x 1 = t, x 2 ≥ s } ) = − λ 1 ( ˜ A ∩ { x : x 1 = t, x 2 = s } ) , whenever A ∩ { x : x 1 = t, x 2 = s } = ∅ or int A ∩ { x : x 1 = t, x 2 = s } 6 = ∅ . These ident ities imply (5.4 ) w ith α = 1. In the gener al case the result follows fro m a reduction to the previo us one via a non-deg enerate linear transforma tion A s uch that A ( L ) = { x : x 1 ≥ 0 , x 2 ≥ 0 } , A ( v 1 ) = (1 , 0 , 0 ) and A ( v 2 ) = (0 , 1 , 0 ). Indeed we have ∂ 2 ∂ s∂ t λ 3 ( ˜ A ∩ ( L + tv 1 + sv 2 )) = | det A| ∂ 2 ∂ s∂ t λ 3 ( A − 1 ( ˜ A ) ∩ { x : x 1 ≥ t, x 2 ≥ s } ) = | det A| λ 1 ( A − 1 ( ˜ A ) ∩ { x : x 1 = t, x 2 = s } ) = | det A| kA (0 , 0 , 1 ) k − 1 λ 1 ( ˜ A ∩ ( R 0 + tv 1 + sv 2 )) .  Lemma 5. 3. As s ume ther e exists i ∈ { 1 , . . . , n } su ch that z i / ∈ F ∩ G . T hen the p olygon, b etwe en F and G , which do es n ot c ontain z i , and the p olygon, b etwe en F ′ and G ′ , which do es not c ontain z i have e qual − 1 -chor d functions at z i . 14 GABRIELE BIANCHI Pr o of. W e pr ov e, for instance, that if z i ∈ ( G \ F ) ∩ ( G ′ \ F ′ ) then F and F ′ hav e the same − 1- chord functions at z i . Let R b e the edg e of B and B ′ with the prop erty that z i is collinear to R , and let l b e the line containing z i and R . The dihedral cones L := cone ( B , R ) and cone ( B ′ , R ) co incide, due to hypothesis (iii) in P rop osition 5 .1. W e claim tha t (5.5) A ∩ ( B + x ) = A ∩ ( L + x ) and A ′ ∩ ( B ′ + x ) = A ′ ∩ ( L + x ) , for ea ch x in a suitable neighbourho o d V of z i . Let π b e any plane through O which strictly s uppo rts H at z i , and let π + be the closed halfspa ce b ounded by π not con taining H . W e hav e A ∩ π + = { O } a nd B ⊂ π + . Since l ⊂ π , we also hav e B + z i ⊂ π + and O ∈ R + z i . These arguments imply A ∩ ( B + z i ) = { O } . Therefore , when x is clos e to z i , we have A ∩ ( B + x ) = A ∩ ( L + x ). Similar arguments prov e A ′ ∩ ( B ′ + x ) = A ′ ∩ ( L + x ). The identities (5 .5) imply λ 3 ( A ∩ ( L + x )) = g A,B ( x ) = g A ′ ,B ′ ( x ) = λ 3 ( A ′ ∩ ( L + x )) , for ea ch x ∈ V . The latter and Lemma 5.2 imply λ 1 ( A ∩ ( l + y )) = λ 1 ( A ′ ∩ ( l + y )) , for all y in a neighbour ho o d of O such that l + y meets int A or do es not meet A , and, mor eov er , l + y meets in t A ′ or do es not meet A ′ . Since the left and the right hand side in the pr evious formula are homo geneous functions o f y of deg r ee 1, and they are co ncav e on their supp orts, the pre v ious ident ity holds for all y , that is, A and A ′ hav e equal X − rays in the directio n of l . The passage from X -rays to − 1-chord functions co mes from [Bia08, Th. 1.3 ], which proves tha t if tw o cones A and A ′ hav e the same X - r ays in the direction of l then their sections F and F ′ with the pla ne { x : x 3 = 1 } hav e the same − 1- chord functions at l ∩ { x : x 3 = 1 } , that is, at z i .  5.2. The set of C 3 discon tin uities of the cov ariogram. Lemma 5.4. L et C ⊂ R 3 b e a dihe dr al c one with e dge the x 1 axis, let D ⊂ R 3 b e a dihe dr al c one with e dge the x 2 axis and assu me that no fac et of C or of D is c ontaine d in { x : x 3 = 0 } . F or t ∈ R , let g ( t ) = λ 3  A ∩ ( D + (0 , 0 , t )) ∩ B (0 , 1)  . Then d 3 g /dt 3 is disc ontinuous at t = 0 . Mor e pr e cisely, if b oth C and D me et b oth { x : x 3 > 0 } and { x : x 3 < 0 } then lim t → 0 + d 3 g dt 3 ( t ) > lim t → 0 − d 3 g dt 3 ( t ) , while if C me ets b oth { x : x 3 > 0 } and { x : x 3 < 0 } and D ⊂ { x : x 3 ≥ 0 } then lim t → 0 + d 3 g dt 3 ( t ) < lim t → 0 − d 3 g dt 3 ( t ) . Pr o of. Assume C ⊂ { x : x 3 ≤ 0 } and D ⊂ { x : x 3 ≥ 0 } . In this ca se C ∩ ( D + (0 , 0 , t )) is empty when t > 0, and it is a tetra hedron of edge lengths prop or tional to | t | when t < 0 and | t | is small. Thus g ( t ) = − αt 3 1 ( −∞ , 0] , for ea ch t in a neighbo urho o d of 0 and for so me α > 0, and we hav e lim t → 0 + d 3 g ( t ) /dt 3 > lim t → 0 − d 3 g ( t ) /dt 3 . Now as s ume that C meets bo th { x : x 3 > 0 } a nd { x : x 3 < 0 } , while D ⊂ { x : x 3 ≥ 0 } . Let C ′ ⊂ { x : x 3 ≤ 0 } b e a clo sed dihedral cone with edge the x 1 axis such that int C ∩ int C ′ = ∅ and C ∪ C ′ is an halfspace π + . Clear ly g ( t ) = λ 3 ( π + ∩ ( D + (0 , 0 , t )) ∩ B (0 , 1 )) − λ 3 ( C ′ ∩ ( D + (0 , 0 , t )) ∩ B (0 , 1)) . THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 15 Since the ﬁrst term in the right hand side of the formula is a C 3 function of t and since the previous case applies to the seco nd ter m, we hav e lim t → 0 + d 3 g ( t ) /dt 3 < lim t → 0 − d 3 g ( t ) /dt 3 . Now a ssume that b oth C a nd D meet b oth { x : x 3 > 0 } and { x : x 3 < 0 } . Let D ′ ⊂ { x : x 3 ≥ 0 } b e a clos ed dihedra l co ne with edge the x 2 axis suc h that int D ∩ int D ′ = ∅ and D ∪ D ′ is an halfspac e . Arguing as ab ove o ne writes g ( t ) as a C 3 function minus λ 3 ( C ∩ ( D ′ + (0 , 0 , t )) ∩ B (0 , 1)). Since the previous case applies to this last function, we ha ve lim t → 0 + d 3 g ( t ) /dt 3 > lim t → 0 − d 3 g ( t ) /dt 3 . When bo th C and D are cont ained in { x : x 3 ≥ 0 } s imila r ideas prov e the claim.  Lemma 5. 5. L et A and B b e c onvex p olyhe dr al c ones in R 3 with non-empty in- terior satisfying (5.1) , let S 3 ( A, B ) = cl  x ∈ R 3 : g A,B fails t o b e C 3 at x  and E ( A, B ) = { R + T : R is an e dge of A and T is an e dge of − B } . Then E ( A, B ) ⊂ S 3 ( A, B ) ⊂ ∂ A ∪ ( − ∂ B ) ∪ E ( A, B ) . Pr o of. Let W ⊂ R 3 \  ∂ A ∪ ( − ∂ B ) ∪ E ( A, B )  be a connected set. When x ∈ W neither the vertex x of B + x belo ngs to ∂ A , nor the vertex O o f A b elong s to ∂ B + x , nor an edg e of A intersects an edg e of B + x . Th us the combinatorial structure of ∂ ( A ∩ ( B + x )) do es not change in W . Since the vertices of A ∩ ( B + x ) are smo oth functions of x , for ea ch x ∈ W , so is g A,B . This prov e s the inclusion S 3 ( A, B ) ⊂ ∂ A ∪ ( − ∂ B ) ∪ E ( A, B ). W e pr ove E ( A, B ) ⊂ S 3 ( A, B ). The s e t E ( A, B ) is cont ained in a ﬁnite set E of planes thr o ugh O whose intersection with E ( A, B ) has dimension 2. F o r π ∈ E let E 0 ( π ) = { R : R ⊂ π is an edg e of A } ∪ { T : T ⊂ π is an edge of − B }∪ ∪ { ( R + T ) ∩ π : R and T are edg e s of A or o f − B not con tained in π } . It suﬃces to pr ove E ( A, B ) \∪ π ∈E E 0 ( π ) ⊂ S 3 ( A, B ), since cl ( E ( A, B ) \∪ π ∈E E 0 ( π )) = E ( A, B ) (b ecause ∪ π ∈E E 0 ( π ) is a ﬁnite union of rays) and S 3 ( A, B ) is closed. Let π ∈ E , x 0 ∈ π ∩ ( E ( A, B ) \ E 0 ( π )) and let v ∈ S 2 be or thogonal to π . Assume that π contains t wo diﬀerent edges R 1 and R 2 of A , t wo diﬀerent edge s T 1 and T 2 of B , and π do es not co ntain a ny facet of A or of B . The choice x 0 ∈ π ∩ E ( A, B ) implies tha t at least tw o of the sets R i ∩ ( T j + x 0 ), i, j = 1 , 2, a re non- empt y . Assume, for instance, that all four sets are no n-empty and let { p i,j } = R i ∩ ( T j + x 0 ). F or ε > 0 suﬃciently small, i , j = 1 , 2 and x in a neighbourho o d o f x 0 , let g 0 ( x ) = λ 3  A ∩ ( B + x ) ∩  R 3 \ ∪ 2 i,j =1 B ( p i,j , ε )   , g i,j ( x ) = λ 3  A ∩ ( B + x ) ∩ B ( p i,j , ε )  . Clearly g A,B = g 0 + P 2 i,j =1 g i,j . The choice x 0 / ∈ E 0 ( π ) implies that x 0 / ∈ ∂ A , O / ∈ ( ∂ B + x 0 ) and that the p oints p i,j are the only intersections o f edges of A with edges of B + x 0 . Th us, arg umen ts similar to those us ed in the ﬁrst pa r t o f the pro o f imply g 0 C 3 in a neig hbourho o d of x 0 . Moreov er, Lemma 5.4 prov e s that P 2 i,j =1 g i,j fails to b e C 3 at x 0 . Similar a r guments prove that g A,B fails to b e C 3 at x 0 when π do es not contain a ny facet of A or of B a nd π contains one or tw o edges o f A and one or tw o edges of B . Now assume that π contains a face t R of A , t wo diﬀerent edges T 1 and T 2 of B , and π do es not contain any facet o f B . The choice x 0 ∈ π ∩ E ( A, B ) implies that at leas t one betw een λ 1 ( R ∩ ( T 1 + x 0 )) a nd λ 1 ( R ∩ ( T 2 + x 0 )) is po sitive. Ass ume, for ins tance, that b oth ter ms ar e p ositive. F o r ε > 0 s uﬃcien tly sma ll, j = 1 , 2 and 16 GABRIELE BIANCHI x in a neigh b ourho o d of x 0 , le t W j = ∪ p ∈ R ∩ ( T j + x 0 ) B ( p, ε ) \ B ( x 0 , ε ), g 0 ( x ) = λ 3  A ∩ ( B + x ) ∩  R 3 \ ( B ( x 0 , ε ) ∪ W 1 ∪ W 2 )   , g j ( x ) = λ 3  A ∩ ( B + x ) ∩ W j  , g 3 ( x ) = λ 3  A ∩ ( B + x ) ∩ B ( x 0 , ε )  . Clearly g A,B = P 3 j =0 g j . The choice x 0 / ∈ E 0 ( π ) implies that O / ∈ ( ∂ B + x 0 ) a nd that the intersections of edges of A with edges o f B + x 0 are contained in W 1 ∪ W 2 . Thu s g 0 is C 3 in a neighbourho o d of x 0 . Arg uing a s in the pro of of Lemma 5.4 prov es the following formulas, v alid for suitable α 1 , α 2 , β > 0:  lim t → 0 + − lim t → 0 −  X j =1 , 2 d 2 g j dt 2 ( x 0 + tv ) = X j =1 , 2 α j  λ 1 ( R ∩ ( T j + x 0 )) − ε  ;  lim t → 0 + − lim t → 0 −  d 2 g 3 dt 2 ( x 0 + tv ) ≥ − εβ . Thu s P 3 j =1 g j fails to b e C 2 at x 0 when ε > 0 is small enough. Similar arg ument s prov e that g A,B fails to be C 2 at x 0 when π contains a facet A and one edg e (but no fac et) of B , and also prove that g A,B fails to be C 1 at x 0 when π contains a facet of A and a facet o f B .  Remark 5.6. The set ∂ A ∪ ( − ∂ B ) is not necessar ily contained in S 3 ( A, B ), even if g enerically it is. F or instanc e , let A = pos c onv { (1 , 1 , 1 ) , (1 , − 1 , 1) , ( − 1 , 1 , 1) , ( − 1 , − 1 , 1) } and let B b e any cone which satisﬁes (5 .1) and with a face T con ta ined in { x : x 2 = 0 } . Then g A,B is C 3 at any point x 0 ∈ r elint T . This cla im relies ultimately on the smo o thness o f the function g ( t ) = λ 3 ( A ∩ { x : x 2 ≥ t, x 3 ≤ 1 } ) = (2 / 3 − t + t 3 / 3)1 [ − 1 , 1] in ( − 1 , 1). 5.3. Pro of of Prop o sition 5.1. W e recall a res ult prov ed in [Bia08]. Lemma 5.7 ([Bia08]) . L et K a nd K ′ b e c onvex p olygons with e qual − 1 -chor d functions at p 1 , p 2 , . . . , p s ∈ R 2 \ K . If conv ( K , p 1 , . . . , p s ) = co nv ( K ′ , p 1 , . . . , p s ) then K = K ′ . Pr o of of The or em 5.1. Within this pr o of we say that z i is neutr al if z i ∈ F ∩ G ∩ F ′ ∩ G ′ , that z i is c onc or dant if z i ∈ ( F \ G ) ∩ ( F ′ \ G ′ ) o r z i ∈ ( G \ F ) ∩ ( G ′ \ F ′ ), and that z i is disc or dant if z i ∈ ( F \ G ) ∩ ( G ′ \ F ′ ) or z i ∈ ( G \ F ) ∩ ( F ′ \ G ′ ). Assumption (i) implies that this cla ssiﬁcation is exhaustive. Claim 5.7. 1. If one vertex of H is c onc or dant then no vertex of H is disc or dant. Pr o of. Lemma 5.5, when expr essed in terms of S 3 ( F, G ) := S 3 ( A, B ) ∩ { x : x 3 = 1 } , implies (5.6) { [ a, b ] : a is a vertex of F and b is a vertex of G } ⊂ S 3 ( F, G ) ⊂ ⊂ ∂ F ∪ ∂ G ∪ { [ a, b ] : a is a vertex o f F and b is a v ertex of G } . Moreov er ana logous inclusions hold for S 3 ( F ′ , G ′ ) := S 3 ( A ′ , B ′ ) ∩ { x : x 3 = 1 } . The iden tity g A,B = g A ′ ,B ′ implies (5.7) S 3 ( F, G ) = S 3 ( F ′ , G ′ ) . Assume the cla im false and that z i is conc o rdant a nd z j is discordant, for some i and j . Ass ume a lso that this means, for instance, z i ∈ ( F \ G ) ∩ ( F ′ \ G ′ ) and z j ∈ ( F \ G ) ∩ ( G ′ \ F ′ ) . THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 17 Hypo thesis (ii) implies that a vertex of H adjacent to a concordant vertex of H is either concorda nt or neutral. Thus there exist h a nd k s uch tha t z h ∈ ( z i , z j ) ∂ H , z k ∈ ( z j , z i ) ∂ H and b oth z h and z k are neutral. Without loss of generality , we may also as s ume z m neutral, whenever z m ∈ ( z i , z j ) ∂ H . W e have [ z i , z j ] ⊂ S 3 ( F ′ , G ′ ), by (5.6) and b e cause z i is a vertex of F ′ and z j is a vertex of G ′ . Since z h ∈ ( z i , z j ) ∂ H and z k ∈ ( z j , z i ) ∂ H are vertices of F a nd of G , relint [ z i , z j ] ∩ ∂ F = ∅ and r e lin t [ z i , z j ] * ∂ G . Therefore (5.6) a nd (5.7) imply that relint [ z i , z j ] must c o ntain a vertex p of G . Since z k ∈ G a nd z m ∈ G , whenever z m ∈ ( z i , z j ) ∂ H , then p is not contained in the co nv ex envelope of these p oints, that is, p is co ntained in int co n v { z k , z i , z i +1 } or in int conv { z k , z j , z j − 1 } . Assume p ∈ in t con v { z k , z i , z i +1 } . W e hav e [ z k , p ] ⊂ S 3 ( F, G ) a nd, by (5.7), [ z k , p ] ⊂ S 3 ( F ′ , G ′ ). Let q be the p oint of relint [ z i , z i +1 ] collinear to z k and p . Since z k is a vertex of F ′ and [ z i , z i +1 ] is an edg e of F ′ , relint [ z k , q ] ∩ ∂ F ′ = ∅ and z k is the only vertex of F ′ contained in [ z k , q ]. Th us [ z k , p ] ⊂ S 3 ( F ′ , G ′ ) a nd (5.6) implies [ z k , p ] ⊂ ∂ G ′ ∪ { [ z k , b ] : b is a vertex of G ′ } . This is p ossible only if [ p, q ] contains a v ertex p ′ of G ′ . Since [ z j , p ] ⊂ conv { z j , z k , p ′ } and z j , z k , p ′ ∈ G ′ , we hav e [ z j , p ] ⊂ G ′ ∩ [ z i , z j ]. On the other hand, s ince p , z h and z k are v ertices of G and z j / ∈ G , we hav e G ∩ [ z i , z j ] ( [ z j , p ]. Ther efore G ∩ [ z i , z j ] ( F ′ ∩ [ z i , z j ] , and the − 1 -chord functions of G and G ′ at z i in the dir ection of z j − z i diﬀer. This violates the conclusio n of Lemma 5.3. When p ∈ in t co nv { z k , z j , z j − 1 } simila r arguments give a contradiction, by proving that the − 1-chord functions o f G and F ′ at z j diﬀer.  T o conclude the pro of it suﬃces to show that either F = F ′ and G = G ′ or else F = G ′ and G = F ′ . If ea ch z i is neutra l then F = F ′ = G = G ′ = H . Now assume tha t no z i is discordant. Let z i 1 , . . . , z i s , for s ome s ≥ 0, b e the v ertices of H which ar e not vertices of F . These p oints a r e also the v ertices of H which are not v ertices of F ′ , b eca use no z i is disco rdant. Since H = conv { F, z i 1 , . . . , z i s } = con v { F ′ , z i 1 , . . . , z i s } , the identit y F = F ′ is a c onsequence of Lemma s 5.3 and 5.7. One pr oves G = G ′ substituting F with G and F ′ with G ′ in the previous ar guments. When no z i is concorda nt a similar pro of g ives F = G ′ and G = F ′ .  6. Determining the suppor t cones of P : proof of (1.4) in Theorem 1.2 The next lemma proves (1.4) in a particular case and it is necessary als o for its pro of in the ge ne r al cas e. The idea b ehind this pro o f is the following: when the cones to be determined are supp ort cones in antipo dal parallel edg es of P of equal length, the pr o blem is substan tially t wo-dimensional and can be reduced to the one studied in Lemma 3.2. Lemma 6. 1. Assume that S 1 := P w and S 2 := P − w ar e p ar al lel e dges of P of e qual length, for some w ∈ S 2 . Then S ′ 1 := P ′ w and S ′ 2 := P ′ − w ar e e dges of P ′ p ar al lel to S 1 whose lengths e qu al λ 1 ( S 1 ) . Mor e over, either co ne ( P, S 1 ) = cone ( P ′ , S ′ 1 ) and cone ( P, S 2 ) = cone ( P ′ , S ′ 2 ) or else cone ( P, S 1 ) = − cone ( P ′ , S ′ 2 ) and cone ( P, S 2 ) = − cone ( P ′ , S ′ 1 ) . Pr o of. F o rmula (1.3) in Theorem 1.2 implies that S ′ 1 or S ′ 2 is an edge o f P ′ parallel to S 1 whose length is λ 1 ( S 1 ). Assume, for instance, that S ′ 2 is such an edg e. Apply again (1.3) with the r oles of P and P ′ exchanged. Either S 1 or S 2 is a translate of S ′ 1 . Thus, also S ′ 1 is an edge of P ′ parallel to S 1 whose le ng th is λ 1 ( S 1 ). 18 GABRIELE BIANCHI Let u ∈ S 2 be the dire ction of S 1 , S 2 , S ′ 1 and S ′ 2 , let D i = cone ( P, S i ), D ′ i = cone ( P ′ , S ′ i ), C i = D i | u ⊥ and C ′ i = D ′ i | u ⊥ , for i = 1 , 2. W e recall that l u denotes the line throug h 0 pa rallel to u . Since D i and D ′ i are dihedra l cones which coincide with C i + l u and C ′ i + l u , resp ectively , the lemma is pr ov ed once w e sho w that either C 1 = C ′ 1 and C 2 = C ′ 2 or else C 1 = − C ′ 2 and C 2 = − C ′ 1 . Let y ∈ R 3 satisﬁes S 1 = S 2 + y . Since conditions (2 .3) and (2.2) imply S 1 − S 2 = S ′ 1 − S ′ 2 , we have S ′ 1 = S ′ 2 + y . If W i denotes a suﬃciently small neighbourho o d of S i then we can write (6.1) P ∩ W i =   ( C i + S i ) ∪ E 1 i  \ E 2 i  ∩ W i , where E 1 i and E 2 i are unions of a ﬁnite num b er of conv ex cones a nd each of these cones is contained in D i + S i , has ap ex an endpo int o f S i and intersects the line aﬀ ( S i ) only in its ap ex. In o rder to compute g P ( y + εx ), for x ∈ u ⊥ ∩ S 2 and ε > 0 small, we write (6.2) P ∩ ( P + y + εx ) = ( P ∩ W 1 ) ∩  ( P ∩ W 2 ) + y + εx  . Simple ele men tary calculations lead to the following formulas, for each i , j = 1 , 2: λ 3  ( C 1 + S 1 ) ∩ ( C 2 + S 2 + y + εx )  = ε 2 λ 2  C 1 ∩ ( C 2 + x )  λ 1 ( S 1 ) ; λ 3  ( C 1 + S 1 ) ∩ ( E i 2 + y + εx )  ≤ O ( ε 3 ) ; λ 3  E i 1 ∩ ( C 2 + S 2 + y + εx )  ≤ O ( ε 3 ) ; λ 3  E j 1 ∩ ( E i 2 + y + εx )  ≤ O ( ε 3 ) . These for m ulas, (6.1) and (6.2) imply g P ( y + εx ) = ε 2 λ 1 ( S 1 ) g C 1 ,C 2 ( x ) + O ( ε 3 ) . The co rresp onding asymptotic expansion for g P ′ is pr ov ed by simila r a rguments. These ex pansions, the identit y g P = g P ′ and the homogeneity of deg ree 2 o f g C 1 ,C 2 and g C ′ 1 ,C ′ 2 imply g C 1 ,C 2 ( x ) = g C ′ 1 ,C ′ 2 ( x ) , for each x ∈ u ⊥ . Observe that C 1 ∩ ( − C 2 ) and C ′ 1 ∩ ( − C ′ 2 ) ar e non-empty (it is an immedia te conseq uence of the conv exity o f P and P ′ ) and apply Lemma 3.2, with A s ubs tituted by C 1 , B by C 2 , A ′ by C ′ 1 and B ′ by C ′ 2 . This lemma implies { C 1 , − C 2 } = { C ′ 1 , − C ′ 2 } , as expla ined in Remark 3.3 .  Now we are r eady to prov e (1.4) in the gener al case. Pr o of of (1.4) in The or em 1.2. W e distinguish three cas es ac cording to dim P w . Case 1. P w is a fac et . In this case (1.4) is an immediate co nsequence of (1.3). Indeed, w e hav e cone ( P, P w ) = cone ( P ′ , P w ) = pos { w } . Case 2. P w is an e dge . By (1.3), we may assume S := P w = P ′ w . W e prov e that if cone ( P , S ) 6 = c o ne ( P ′ , S ) then, for a suita ble y ∈ R 3 , − S + y is an edge o f P ′ and (6.3) cone ( P, S ) = − co ne ( P ′ , − S + y ) . This clearly implies cone ( P , S ) = cone ( − P ′ + y , S ) and P w = ( − P ′ + y ) w . Thus, (1.3) and (1.4) hold with − P ′ + y repla cing P ′ . When cone ( P, S ) 6 = cone ( P ′ , S ) w e may assume, without loss of g enerality , the e xistence of a face t R of P containing S such that aﬀ ( R ) supp orts P ′ and intersects ∂ P ′ only in S . Let w 1 be the unit outer norma l to P at R . W e hav e P w 1 = R and P ′ w 1 = S and, therefore, P ′ w 1 is not a tra nslate of P w 1 . F or mu la (1.3) in Theo rem 1.2, with w substituted by w 1 , implies that ( − P ′ ) w 1 + y = R , for some y ∈ R 3 , that is, − R + y is a facet of P ′ with outer norma l − w 1 . In par ticular, since S is an edge o f R , − S + y is THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 19 an edg e of P ′ . Since − w 1 ∈ N ( P ′ , − S + y ) and w 1 ∈ int N ( P ′ , S ), S and − S + y are antipo dal paralle l edges o f P ′ of equal length. Lemma 6.1 implies (6.3), since cone ( P, S ) 6 = cone ( P ′ , S ). Case 3: P w is a vertex . Up to a sma ll p erturbation of w , we may as s ume that also P − w is a vertex. Note that the v alidity of (1.4) for the p er tur be d w implies its v alidity fo r the origina l w to o . Conditions (2.2) and (2.3) imply P w − P − w = P ′ w − P ′ − w . Therefo r e, P ′ w − P ′ − w consists of one element, and this may happen only if b oth P ′ w and P ′ − w are p oints. Up to a translation of P ′ and an aﬃne transformatio n, we may assume P w = P ′ w = (0 , 0 , − 1), P − w = P ′ − w = (0 , 0 , 1 ), (6.4) P ∪ P ′ ⊂ { x : − 1 ≤ x 3 ≤ 1 } , P ∩ { x : x 3 = − 1 } = P ′ ∩ { x : x 3 = − 1 } = { (0 , 0 , − 1) } , P ∩ { x : x 3 = 1 } = P ′ ∩ { x : x 3 = 1 } = { (0 , 0 , 1 ) } . Let A = c one ( P, (0 , 0 , − 1)), B = cone ( P , (0 , 0 , 1)), A ′ = cone ( P ′ , (0 , 0 , − 1)) a nd B ′ = cone ( P ′ , (0 , 0 , 1)). These cones sa tisfy (5.1). Since we have g A,B ( x ) = g P ( x − (0 , 0 , 2)) = g P ′ ( x − (0 , 0 , 2 )) = g A ′ ,B ′ ( x ) , for each x in a neighbourho o d of O , and s ince g A,B ( x ) and g A ′ ,B ′ ( x ) ar e homoge- neous functions of degree 3 , we hav e g A,B = g A ′ ,B ′ . Let F , G , F ′ , G ′ and H b e deﬁned as in (5.2) a nd (5.3). W e prove that the part of Theorem 1.2 e x pressed by formula (1.3) implies that these sets sa tisfy the assumptions of Pr op osition 5.1 . O nce this is done, Pr op osition 5.1 implies that (1.4) holds , p o ssibly a fter a substitution of P ′ with − P ′ . Hyp othesis (iii) is satisﬁe d. It suﬃces to show this when (6.5) z i ∈ ( F \ G ) ∩ ( F ′ \ G ′ ) , since in the other ca ses the pro of is similar. Condition (6.5 ), when rephras ed in terms of the resp ective co nes, states that p os ( z i ) is an e dge of A , of A ′ (and of D := conv ( A ∪ ( − B ))) whic h mee ts − B and − B ′ only at O . Let w 1 ∈ S 2 ∩ int N ( D , pos ( z i )). When rephras ed in terms of P and P ′ , (6.5) implies tha t (6.6) ( P ′ ) − w 1 is a p oint (it coincides with (0 , 0 , 1)) and that S := P w 1 and S ′ := P ′ w 1 are edges o f P a nd P ′ , resp ectively , with endpo int (0 , 0 , − 1). Let us prov e S = S ′ . Ass ume the contrary . F ormula (1.3) in Theorem 1 .2, with w subs tituted by w 1 , implies that S is a tra nslate o f ( − P ′ ) w 1 (beca use S ′ is not a translate of S ). This contradicts (6.6) a nd prov es S = S ′ . The co incidence of F and F ′ in a neighbour ho o d of z i is clea rly equiv a lent to the identit y (6.7) cone ( P, S ) = cone ( P ′ , S ) . Assume that this ide ntit y is false. Arg uing as in the pro of of Case 1, o ne shows that − S + y is a n edge of P ′ and (6.3) holds, for a suitable y ∈ R 3 . The identit y (6.3) imply ( P ′ ) − w 1 = − S ′ + y (b ecause w 1 ∈ cone ( P, S )), contradicts (6 .6) and prov es (6.7). Hyp othesis (i) is satisﬁe d. Assume z i ∈ F ∩ G . In this ca se p os ( z i ) is an edge of A , of − B and of D . If w 1 ∈ S 2 ∩ in t N ( D , pos ( z i )), the latter implies tha t P w 1 and P − w 1 are antipo dal parallel edges of P with endp oints (0 , 0 , − 1) and (0 , 0 , 1), resp ectively . Arguing as in the ﬁrst lines of the pro of of Lemma 6.1, one shows that also P ′ w 1 and P ′ − w 1 are edges parallel to P w 1 . 20 GABRIELE BIANCHI Let π denote the plane orthogona l to w 1 containing p os ( z i ). Since π supp orts D = conv ( A ∪ B ) = conv ( A ′ ∪ B ′ ) at O , π + (0 , 0 , − 1) s upp or ts b oth P and P ′ at (0 , 0 , − 1), and π + (0 , 0 , 1) supp orts b oth P and P ′ at (0 , 0 , 1). In pa rticular, we hav e w 1 ∈ N ( P ′ , (0 , 0 , − 1)) and − w 1 ∈ N ( P ′ , (0 , 0 , 1)). The latter and (6.4) imply that (0 , 0 , − 1) and (0 , 0 , 1) are endpoints of P ′ w 1 and P ′ − w 1 , resp ectively . When rephrased in ter ms of the suppo rt cones, this implies that p os ( z i ) is an edge of A ′ and of − B ′ or, equiv alently , that z i is a vertex of F ′ and G ′ . Hyp othesis (ii) is satisﬁe d. Assume that [ z i , z i +1 ] is an edge o f F . In this case p os ( z i ) + p os ( z i +1 ) is a facet of A and of D . If w 1 ∈ S 2 ∩ N ( D, pos ( z i ) + po s ( z i +1 )), then P w 1 is a facet of P with v ertex (0 , 0 , − 1 ) and pos ( P w 1 + (0 , 0 , 1)) = po s ( z i )+ po s ( z i +1 ). Arguing a s w e hav e done to prov e that hypothesis (i) is sa tisﬁed shows that (0 , 0 , − 1) is a vertex o f P ′ w 1 and (0 , 0 , 1) is a vertex of P ′ − w 1 . This and formula (1.3) in Theorem 1.2 imply that either P w 1 = P ′ w 1 or − P w 1 = ( − P ′ ) w 1 . In the ﬁr st cas e [ z i , z i +1 ] is an edge o f F ′ , in the second case it is a n edge o f G ′ .  7. The s tructure of synisothetic pol ytopes If P is a translation o r a reﬂection of P ′ then ( P , − P ) is syniso thetic to ( P ′ , − P ′ ), but the conv erse implication is false. Example 7.1. Let P ⊂ R 3 be a conv ex p o lytop e such that ∂ P co n tains a simple closed curve Γ together with − Γ, with Γ ∩ ( − Γ ) = ∅ . The unio n Γ ∪ ( − Γ) disconnects ∂ P in three comp onents. L et Σ 1 be the one b ounded by Γ, Σ 2 the one b ounded by − Γ and Σ 3 the one bo unded by Γ ∪ − Γ. Choo s e P in such a wa y that Σ 1 6 = − Σ 2 , Σ 3 6 = − Σ 3 , ∂ P and − ∂ P coincide in a neig h b ourho o d W of Γ and W contains all face s which intersect Γ. Let P ′ be the p o ly top e whose b oundar y is ( − Σ 1 ) ∪ Σ 3 ∪ ( − Σ 2 ). The p o ly top e P ′ is not a translate or a r eﬂection of P but ( P , − P ) is synisothetic to ( P ′ , − P ′ ). In orde r to in tr o duce some notations, we need the fo llowing t wo lemmas. Lemma 7.2. Le t P and P ′ b e c onvex p olytop es in R 3 with non-empty interior such that ( P , − P ) and ( P ′ , − P ′ ) ar e synisothetic. Then DP = D P ′ . Pr o of. The second iden tit y o f (2 .2 ) implies (7.1) ( D P ) w = P w + ( − P ) w and ( D P ′ ) w = P ′ w + ( − P ′ ) w . The synisothesis of the tw o pairs implies that each summand in the r ig ht hand side of the ﬁrst iden tit y a lso app ear s in the right ha nd side of the second ident ity and vice versa. As a c o nsequence we hav e λ 2 (( DP ) w ) = λ 2 (( DP ′ ) w ) for ea ch w ∈ S 2 . Therefore D P and DP ′ hav e the same 2-a r ea measure and, by the uniqueness assertion for the Minko w s ki P r oblem [Sch93, Th. 7.2.1], they a re tr anslates of each other. Since b o th diﬀer ence bo dies are origin symmetric, they coincide.  In this section P and P ′ will alw ays b e as in the statemen t of Lemma 7.2. Lemma 7. 3. F or e ach w ∈ S 2 ther e exist σ ∈ {− 1 , 1 } and x = x ( σ ) ∈ R 3 such that P w = ( σP ′ ) w + x and cone ( P, P w ) = cone ( σ P ′ , ( σ P ′ ) w ) ; (7.2) P − w = ( σ P ′ ) − w + x and cone ( P , P − w ) = cone ( σ P ′ , ( σ P ′ ) − w ) . (7.3) Pr o of. Condition (7.2) ex plic itly expre s ses the isothesis of P w and ( σ P ) w , for some σ ∈ { − 1 , 1 } , and this holds b y assumption. What w e ha ve to prove is that P − w is isothetic to ( σ P ′ ) − w (with the s ame σ ) and that the translatio n that ca r ries ( σP ′ ) w int o P w also car ries ( σ P ′ ) − w int o P − w . W e know that ( σP ′ ) − w is isothetic either to P − w or to ( − P ) − w . In the ﬁrst case the pro o f reg a rding σ is concluded. Assume that the se cond p ossibility holds. This THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 21 prop erty is equiv a lent to the isothesis of P w and ( − σ P ′ ) w , since for e ach co nv ex po lytop e Q we hav e ( − Q ) − w = − Q w and co ne ( − Q, ( − Q ) − w ) = − cone ( Q , Q w ) . This and (7.2) imply that P w is isothetic b oth to ( σ P ′ ) w and to ( − σ P ′ ) w . Since isothesis is a tra nsitive proper t y , ( σ P ′ ) w and ( − σ P ′ ) w are iso thetic and in (7.3) we can cho ose b oth σ = 1 a nd σ = − 1. Lemma 7 .2 and (7.1) imply P w − P − w = P ′ w − P ′ − w , and this iden tity implies that the trans la tion vector x in (7.3) coincide with the one in (7.2).  Deﬁnition 7.4. Let F b e a prop er face o f P a nd w ∈ S 2 be such that F = P w . W e say that F is p ositive when (7.2) ho lds only with σ = 1; that F is ne gative when (7.2) holds o nly with σ = − 1 ; that F is neutr al when (7.2) holds b oth with σ = 1 and σ = − 1. When F is p ositive or neutral x + ( F ) denotes the transla tion x whic h app ears in (7.2) when σ = 1. When F is negative o r neutral the vector x which appea rs in (7.2) when σ = − 1 is denoted by x − ( F ). The symbol P ′ F denotes P ′ + x + ( F ) when F is p ositive and denotes − P ′ + x − ( F ) when F is negative. It is easy to chec k that the deﬁnition do es not dep end on the choice o f w . Observe that the p olytop e P ′ F is a trans la tion or reﬂection of P ′ with the prop er t y that F is a face of P ′ F and cone ( P , F ) = cone ( P ′ F , F ). Within the rest of this section the terms b oundary , interior a nd neigh b o urho o d of a subset of ∂ P alwa ys r efer to the r elative top olog y induced on ∂ P by its immer - sion in R 3 , with the exception of ∂ P and ∂ P ′ which keep their orig inal meaning . Moreov er the term face will a lwa ys mean prop er face. Let us try to express the global aim o f the lemmas in this section in terms which a re the least technical po ssible. Let G b e any fa ce o f P which is no t neutral. W e pr ov e that there is a subset Σ + of ∂ P ∩ ∂ P ′ G , which contains G , to whic h it cor resp onds an “antipo da l” subset Σ − of ∂ P ∩ ∂ P ′ G . Moreover, if G and y ∈ R 3 are suitably chosen then the bo undaries of Σ + and of − Σ − + y co incide and ∂ P , ∂ P ′ G , − ∂ P + y a nd − ∂ P ′ G + y all coincide in an one-sided neighbour ho o d of ∂ Σ + . The set Σ + is the comp one nt of the intersection of ∂ P (with so me exceptional faces remov ed) a nd ∂ P G which contains G . Deﬁnition 7.5. Let G 0 be a p ositive fac e of P . Let F b e the colle c tion o f the edges or facets F of P which are po sitive or neutra l and satisfy x + ( F ) 6 = x + ( G 0 ). Let Σ = in t  ∂ P ′ G 0 \ ∂ P \ [ F ∈F F  and let Σ + be the clo sure o f the comp onent of Σ which contains relint G 0 . If G 0 is nega tive we deﬁne F , Σ and Σ + as a bove, with po sitive s ubstituted b y nega tive and x + substituted by x − . In the prev ious deﬁnition we hav e implicitly used the inclusion r e lin t G 0 ⊂ Σ, which is proved in next lemma. Lemma 7.6. We have relint G 0 ⊂ Σ ∩ in t Σ + . Pr o of. It suﬃces to prove relint G 0 ⊂ Σ, since this inclusion implies immedia tely relint G 0 ⊂ int Σ + . When G 0 is a facet the inclusion is obvious. Now assume that G 0 is a po sitive vertex q . Since co ne ( P , q ) = cone ( P ′ G 0 , q ), P and P ′ G 0 coincide in a neighbourho o d of q . Therefor e to each fac e F o f P containing q it c orresp o nds a fa c e F ′ of P ′ G 0 containing q with cone ( P , F ) = cone ( P ′ G 0 , F ′ ). If F is p ositive or neutral then it neces sarily co incides with F ′ (a convex p oly to pe has an unique face with a given suppo rt cone) and ther efore x + ( G ) = x + ( q ). This prov es that no face of P containing q b elo ngs to F , a nd this implies that a neighbourho o d of 22 GABRIELE BIANCHI q is contained in Σ. Similar arguments prov e relint G 0 ⊂ Σ when G 0 is a negative vertex or an edge.  The next lemma pr ov es that, when P is not a translatio n o r r eﬂection of P ′ , P has b oth p ositive and negative facets (and th us Σ + 6 = ∂ P ). Lemma 7.7. If n o fac et of P is ne gative (is p ositive) then P ′ is a tr anslate of P (of − P , r esp e ctively). If e ach fac et of P is neutr al then P and P ′ ar e c entr al ly symmetric. Pr o of. Assume that no facet of P is neg ative. Let F 1 and F 2 be adjacent facets of P (in the sense that they hav e an edge S in commo n). W e prov e that x + ( F 1 ) = x + ( F 2 ). Indeed, if x + ( F 1 ) 6 = x + ( F 2 ) then F 2 is not a face of P ′ + x + ( F 1 ) and the facet of P ′ + x + ( F 1 ) c ontaining S and diﬀerent from F 1 has no corr e sp o nding facet in P . Since each facet o f P can be joined to F 1 by a ﬁnite s equence o f adjacent facets, we hav e P = P ′ + x + ( F 1 ). A simila r argument proves the claim reg arding the absence of pos itive facets. These tw o implications together show tha t when P has only neutra l facets b oth P and P ′ are cen trally symmetric.  Standard ar guments o f gener a l top ology prov e that Σ + , the clo sure of a con- nected op en set, coincides with cl in t Σ + . W e recall that for an o p en subset of R 2 (and thus a lso for an op en s ubs et of ∂ P ) b eing connected is equiv a le nt to b eing path-connected. The set ∂ Σ + is clea rly the union of ﬁnitely ma n y s egments, which we ca ll edges. Lemma 7 . 8. Assum e that P is not a tr anslation or a r eﬂe ction of P ′ , t hat G 0 is p ositive and x + ( G 0 ) = 0 . L et S b e an e dge of ∂ Σ + . The fol lowing assertions hold. (i) Ther e exist a fac et F of P and a fac et F ′ of P ′ , with F / ∈ F and in t F ∩ F ′ 6 = ∅ , such that F ∩ F ′ = F ∩ Σ + and S is an e dge of the p olygon F ∩ F ′ . (ii) Ther e exists y = y ( S ) ∈ R 3 such that the fol lowing pr op erties hold: (a) If S is an e dge of P and of P ′ , t hen it is an e dge of − P + y and of − P ′ + y to o and (7.4) cone ( P, S ) = cone ( − P ′ + y , S ) 6 = co ne ( P ′ , S ) = cone ( − P + y , S ) . In this c ase S is ne gative and y = x − ( S ) . (b) If S is not an e dge of P or it is not an e dge of P ′ , then F and F ′ ar e also fac ets r esp e ctively of − P ′ + y and − P + y . In this c ase F is ne gative and y = x − ( F ) . (iii) Ther e exist a n eighb ourho o d W of relint S su ch that (7.5) Σ + ∩ W ⊂ ( − ∂ P + y ) ∩ ( − ∂ P ′ + y ) . Pr o of. Claim (i). Let z ∈ r elint S . Ther e exists a sequence ( z n ) of p oints of ∂ P conv er g ing to z a nd contained in the component of Σ containing relin t G 0 . W e may assume that inﬁnitely many terms of this sequence are contained in int ( F ∩ F ′ ), where F and F ′ are suitable coplanar facets of P and P ′ , res p ectively , cont aining S a nd with F / ∈ F . Since int ( F ∩ F ′ ) is contained in Σ and contains p oints path-connected to r elint G 0 by a path in Σ, it is contained in the comp onent of Σ containing relint G 0 . Thus F ∩ F ′ ⊂ Σ + . On the other hand, we hav e F ∩ Σ + ⊂ F ∩ ∂ P ′ (b y deﬁnition of Σ + ) a nd F ∩ ∂ P ′ = F ∩ F ′ (b y the c o nv exity of P and P ′ ). All these inc lus ions imply (7.6) F ∩ F ′ = Σ + ∩ F . In view o f (7 .6) and S ⊂ ∂ Σ + , S cannot int ersect int ( F ∩ F ′ ). Thus S is contained in an edge of the poly gon F ∩ F ′ . It is eas y to pr ov e that it co incides with such an edge. THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 23 Claim (ii). W e assume that (7.7) S is an edge of P and P ′ and prove that (7.8) cone ( P, S ) 6 = cone ( P ′ , S ) . Assume (7.8) fa lse and let us prov e that Σ + “extends on bo th sides of S ”. Let G be the facet of P containing S and diﬀerent from F , and let G ′ be the facet of P ′ containing S and diﬀerent from F ′ . If (7.8) is false, G and G ′ are coplanar and their intersection has non-empt y interior and is path-connected to S . W e pr ov e tha t S / ∈ F and G / ∈ F . The a s sumption (7.7) and the denial of (7.8) imply S p ositive or neutral a nd x + ( S ) = 0. Thus, S / ∈ F . Assume that G is po sitive or neutral. In this c ase b oth G + x + ( G ) and G ′ are facets of P ′ with the same outer no r mal and th us they coincide. Since G and G ′ share the edg e S , it has to b e x + ( G ) = 0. This prov es G / ∈ F . Since S / ∈ F and G / ∈ F , we hav e ( F ∩ F ′ ) ∪ S ∪ ( G ∩ G ′ ) ⊂ Σ + . Thu s S ∩ int Σ + 6 = ∅ and S is not con tained in ∂ Σ + . Assume again (7 .7). Condition (7.8) implies that S is nega tive. Cho os e w ∈ S 2 so that S = P w and deﬁne y = x − ( S ). Conditions (7.2) and (7.3) imply that S is a co mmon edge of − P + y and − P ′ + y and (7.4 ) ho lds. Now assume (7.7) false. In this case we have F 6 = F ′ , which implies that F is negative. In fa ct, if F is p ositive or neutral then x + ( F ) 6 = 0, b ecause F is not a facet of P ′ , but x + ( F ) 6 = 0 co n tradicts F / ∈ F , since x + ( G 0 ) = 0. Cho ose w ∈ S 2 so that F = P w and deﬁne y = x − ( F ). Conditions (7.2) a nd (7 .3) imply that F is a facet of − P ′ + y and F ′ is a facet of − P + y . Claim (iii) is an immediate consequence of Claims (i) and (ii ).  The num b er of comp onents of ∂ Σ + is ﬁnite since the n umber of edges of ∂ Σ + is ﬁnite. Each comp onent is a p olyg onal curve. Lemma 7. 9 . Assume that P is not a tr anslation or a r eﬂe ction of P ′ , that G 0 is p ositive and x + ( G 0 ) = 0 . L et C 1 , . . . , C s , for a su itable s > 0 , denote the c omp onents of ∂ Σ + and let m ∈ { 1 , . . . , s } . The ve ctor y asso ciate d to e ach e dge of C m by L emma 7.8 do es not dep end on the e dge and (7.5) holds true when W is a suitable neighb ourho o d of C m . Pr o of. Let S 1 and S 2 be any tw o adjacent edg e s of C m and let q b e the common endpo int . F or each i = 1 , 2, let F i , F ′ i and y ( S i ) be the facets a nd v ector asso cia ted to S i by Lemma 7.8. T o pr ov e the lemma it suﬃces to prov e that y ( S 1 ) = y ( S 2 ) and that (7.9) (7.5) holds when W is a suitable neighbourho o d o f q , since any tw o edge s of C m are joined b y a ﬁnite sequence of adjacent e dg es of C m . W e divide the pro of in three c a ses. In the ﬁrst tw o case s the cla im follows from the existence o f a s uitable face containing q whos e vector x − coincides bo th with y ( S 1 ) and y ( S 2 ). In the thir d case some topolo gical ar guments are needed. Case 1. The p oint q is a vertex of P and of P ′ . Assume cone ( P , q ) = cone ( P ′ , q ). In this case q is p ositive o r neutr al and a rgu- men ts simila r to those used in the pro of of Lemma 7 .6 show that a neig hbo urho o d of q is contained in Σ + . This is imp os s ible b ecause S 1 , S 2 ⊂ ∂ Σ + . W e may thus a ssume co ne ( P, q ) 6 = co ne ( P ′ , q ). This implies that q is neg ative and that P a nd − P ′ + x − ( q ) co incide in a neighbour ho o d of the common vertex q . If, for so me i , S i is not an edge of P or it is not an edge of P ′ , then Lemma 7.8 (iib) applies and F i is a fa cet o f − P ′ + y ( S i ). This implies y ( S i ) = x − ( q ). Similar arguments prove the s ame iden tity when S i is an edge of P a nd o f P ′ and Lemma 7.8 (iia) applies. In each ca s e w e hav e y ( S 1 ) = x − ( q ) = y ( S 2 ). Claim (7.9) follows from 24 GABRIELE BIANCHI F 1 S ′ 2 F 2 S 1 S 2 F ′ 1 F ′ 2 R R ′ Σ + q t S ′ 1 Figure 3. The set Σ + (in gray) in a neigh b our ho o d of [ q, t ]. Solid lines repre s ent edges of ∂ P , while dotted lines re pr esent edges of ∂ P ′ . the inclusion Σ + ⊂ ∂ P ∩ ∂ P ′ and fro m the coincidence of P and − P ′ + x − ( q ) and of P ′ and − P + x − ( q ) in a neighbourho o d of q . Case 2. The p oint q is in the interior of a fac et of P or of a fac et of P ′ . Assume, for instance, that q is in the interior of a face t of P . In this case F 1 = F 2 and, since F ′ 1 and F ′ 2 are coplanar with F 1 , we hav e F ′ 1 = F ′ 2 to o. Since neither S 1 nor S 2 are edges of P , Lemma 7.8 (iib) applies b oth to S 1 and S 2 . Lemma 7.8 implies the equality y ( S 1 ) = y ( S 2 ) = x − ( F ) and it also implies (7.9 ). Case 3. The p oint q is in t he r elative interior of an e dge (but of no fac et) of P or of P ′ . Assume, for instance, that q is in the rela tive interior o f a n edge R of P . First we obser ve tha t R is b oth an edge of F 1 and of F 2 , be c ause q belong s b oth to F 1 and to F 2 . In addition, neither S 1 nor S 2 can b e edg es of P . W e may assume F 1 6 = F 2 , b ecause when F 1 = F 2 the pr o of can be concluded as in Case 2. W e may also assume R p ositive, b ecause when R is negative or neutral the lemma can be prov ed a s in the second pa r agra ph of Case 1, with R pla ying the role of q . W e claim that the p oint q do e s not belo ng to the relative interior o f an edge of P ′ . If it do es and R ′ is the edge , then R ′ is an edge o f F ′ 1 and F ′ 2 (it is proved as ab ov e) and it is collinear to R (b ecause F i is coplanar to F ′ i , for each i = 1 , 2 ). In this ca se q b elong s to the relative int erior of the edge R ∩ R ′ of F 1 ∩ F ′ 1 . This contradicts the fact, prov ed in Lemma 7.8, that S 1 is an edge of F 1 ∩ F ′ 1 . Let R ′ = R + x + ( R ). See Fig . 3. Obs erve that R ′ is the edge of P ′ in common to F ′ 1 and F ′ 2 , be cause the facets of P ′ which contain R ′ hav e outer normals equal to those of F 1 and F 2 , a nd F ′ 1 and F ′ 2 are the only facets of P ′ with this pr op erty . Moreov er R and R ′ are collinear and x + ( R ) 6 = 0. Thus, R ∈ F . Let R ∩ R ′ = [ q , t ]. F or i = 1 , 2, let S ′ i be the edge o f F i ∩ F ′ i which con tains t and diﬀers fr om [ q , t ]. It is evident that ∂ P diﬀers from ∂ P ′ on one side of S ′ i and that these se gments a re contained in ∂ Σ + . Let C k be the comp onent of ∂ Σ + which contains S ′ 1 and S ′ 2 , and let us prov e that k 6 = m . In a neigh b ourho o d of [ q , t ] the set Σ + coincides with ( F 1 ∩ F ′ 1 ) ∪ ( F 2 ∩ F ′ 2 ) and it is bo unded on one side by S 1 ∪ S 2 and o n the other side by S ′ 1 ∪ S ′ 2 . By deﬁnition of Σ + , for each i = 1 , 2, there is a simple c ontin uous curve Γ i contained in Σ a nd connecting a given po int p i ∈ in t ( F i ∩ F ′ i ) to a given p oint in relint G 0 . Since R ∈ F , we have Γ i ∩ R = ∅ . The union of Γ 1 , Γ 2 and a contin uous path THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 25 connecting p 1 to p 2 and contained in int ( F 1 ∩ F ′ 1 ) ∪ R ∪ int ( F 2 ∩ F ′ 2 ) disconnects ∂ Σ + in tw o s ets, with C m in one set a nd C k in the other s et. This implies k 6 = m . W e say tha t t wo diﬀerent co mp onents C h and C j of ∂ Σ + are F − c onne ct e d by a se gment [ a, b ] if a ∈ C h , b ∈ C j , [ a, b ] is co nt ained in an edge of P whic h b elo ngs to F and ( a, b ) ⊂ int Σ + . What we ha ve prov ed s o far implies that if y ( S 1 ) 6 = y ( S 2 ) then C m is F − c o nnected to another comp onent of ∂ Σ + by a segment with endp oint q . W e prove that any t wo diﬀerent c omp onents C h and C j can b e F − connected b y at most one seg ment. Assume that th ey are F − connec ted by tw o diﬀ erent segments [ a, b ] and [ a ′ , b ′ ], and consider a simple closed curve Γ with Γ ⊂ C h ∪ C j ∪ [ a, b ] ∪ [ a ′ , b ′ ] , and [ a, b ] ∪ [ a ′ , b ′ ] ⊂ Γ . This curve is not contained in Σ a nd it disconnects ∂ P in tw o o pen no n-empty sets such that one of them do es not intersect Σ + . This contradicts the inclusio n ( a, b ) ∪ ( a ′ , b ′ ) ⊂ in t Σ + . A similar ar g ument prov es that ther e is no ﬁnite seq ue nc e i 1 , . . . , i p , where p > 2 and i l 6 = i j whenever l 6 = j , such that C i p is F − co nnected to C i 1 and C i l is F − connected to C i l +1 , for each l = 1 , . . . , p − 1. W e call such a conﬁguratio n a close d cir cuit of F − c onne cte d c omp onents . O bserve that (7.10) y ( S 1 ) − y ( S 2 ) = x − ( F 1 ) − x − ( F 2 ) = y ( S ′ 1 ) − y ( S ′ 2 ) . The corresp onding prop erty holds for any pair of comp onents F -co nnec ted by [ d, e ]: the diﬀerence b etw een the vectors y of C h across d equals the diﬀerence b etw e e n the vectors y of C k across e . W e conclude the pro of of the lemma. Put i 1 = m a nd i 2 = k . W e prove that if C i 2 is F − co nnected only to C i 1 then (7.11) y ( S 1 ) = y ( S 2 ) . In this case there is a sequence R 1 , . . . , R p of diﬀerent co nsecutive edges of C i 2 such that S ′ 1 = R 1 and S ′ 2 = R p . W e have y ( R i ) = y ( R i +1 ) for each i , b ecause C i 2 is F − connected only to C i 1 and the endpoint in common to R i and R i +1 is diﬀerent from t . This implies y ( S ′ 1 ) = y ( S ′ 2 ) a nd (7.11). Similar a rguments prov e (7.11) when there is a co mpo nent C i 3 diﬀerent from C i 1 which is F − connected only to C i 2 . Ther efore, when y ( S 1 ) 6 = y ( S 2 ) it is p ossible to deﬁne an inﬁnite sequence i l , l = 1 , 2 , . . . such that C i l is F − connected to C i l − 1 . Ea ch index in the seque nc e is diﬀerent from all the pr evious ones , be c ause if i l = i p , for s ome l a nd p with p ≤ l , then ther e is a closed circuit of F − connected comp onents. This constructio n contradicts the ﬁniteness of the num b er of co mpo nent s o f ∂ Σ + . Claim (7.9 ) f ollows by Lemma 7.8(iib) and the observ ations con tained in the ﬁrst paragr aphs of Case 3 (see als o Fig. 3).  Lemma 7. 10. L et Λ ⊂ ∂ P have c onne cte d interior and satisfy Λ = cl int Λ , Λ 6 = ∅ and Λ 6 = ∂ P . As s u me that ∂ Λ is the union of ﬁnitely many p olygonal curves (e ach with ﬁnitely many e dges). The b oundary of e ach c omp onent of ∂ P \ Λ is a close d simple curve. Pr o of. Let A be a compo nent of ∂ P \ Λ . Clearly w e have ∂ A ⊂ ∂ Λ . Let us pr ove that ∂ P \ cl A is connected. It s uﬃces to pr ov e tha t each po int q ∈ ∂ P \ cl A is path-connected to int Λ b y a contin uous curve contained in ∂ P \ cl A . If q ∈ ∂ Λ \ cl A this is obvious due to the simple structur e of ∂ Λ. If q b elongs to some comp onent A ′ of ∂ P \ Λ, with A ′ 6 = A , there is a curve contained in A ′ connecting q to so me p oint of ∂ Λ \ cl A . This curve can b e extended to a curve contained in int Λ ∪ A ′ ∪ { q } connecting q to int Λ. Consider ∂ Λ as a graph. If [ a 1 , a 2 ] denotes a n edge of ∂ Λ then there is a t least another edge of ∂ Λ diﬀerent from [ a 1 , a 2 ] and with endp oint a 2 , b ecause otherwise 26 GABRIELE BIANCHI Λ 6 = cl in t Λ. If in addition [ a 1 , a 2 ] is co ntained in ∂ A , then one of the edges of ∂ Λ with endp o int a 2 and diﬀerent from [ a 1 , a 2 ], say [ a 2 , a 3 ], is necess a rily an edge of ∂ A . The iter a tion of this construction deﬁnes a closed simple curve γ ⊂ ∂ A . Let Γ 1 and Γ 2 be the compo nents of ∂ P \ γ . Since A is connected a nd do es not meet γ we have either A ⊂ Γ 1 or A ⊂ Γ 2 . Assume A ⊂ Γ 1 . Since ∂ P \ cl A is connected, contains Γ 2 and doe s not meet γ , w e hav e ∂ P \ cl A ⊂ Γ 2 , that is , cl A ⊃ cl Γ 1 . The previous inclusions imply cl A = cl Γ 1 . Since A = int cl A (it is an easy cons e- quence of the ident ity Λ = cl int Λ) and Γ 1 = in t cl Γ 1 (it follows from the deﬁnition of Γ 1 ) the identit y cl A = cl Γ 1 implies A = Γ 1 . Therefor e ∂ A = γ .  Remark 7.11. This lemma, which seems an inv erse form of J o rdan Curve Theo- rem, does not seem to b e a v a ilable in the literatur e. Andreas Zastrow told us that it can b e derived from Alexander Duality in Alge braic T o po logy; see [Gre67, p. 179]. It can b e pr ov ed that this duality implies that the rank of the ﬁrst homolo gy group H 1 ( ∂ A ) o f ∂ A , (with ∂ A tho ug ht as a graph e mbedded in ∂ P ) eq ua ls the n um b er of the comp onents of ∂ P \ ∂ A minus 1. Since, as pr ov ed ab ove, ∂ P \ ∂ A ha s tw o comp onents, the rank of H 1 ( ∂ A ) is 1. This fact and the pr op erty A = in t c l A imply that ∂ A is a simple closed c urve. Lemma 7. 12. Assume that P is not a tr anslation or a r eﬂe ct ion of P ′ and, for e ach m = 1 , . . . , s , let y m denote the ve ctor asso ciate d t o C m by L emma 7.9. It is p ossible to cho ose G 0 so that y i = y j for e ach i, j = 1 , . . . , s . Pr o of. Given a p ositive or negative fa c e G of P , let F ( G ), Σ( G ) and Σ + ( G ) b e as in Deﬁnition 7.5, with G substituting G 0 . W e asso ciate to G a p ositive in teger s ( G ) as follows. Given B ⊂ ∂ P deﬁne size ( B ) = n umber of fac ets of P whose in ter ior in ter sects B and sz ( G ) = inf A comp onent of ∂ P \ Σ + ( G ) size ( ∂ P \ A ) . Let G 0 denote a face which minimises sz ( G ) ov er a ll positive or neg ative facets of P . Let C 1 , . . . , C s be the comp onents of ∂ Σ + ( G 0 ) and let y i be the vector a sso ciated by Lemma 7.9 to C i . W e claim that y i = y j for ea ch i, j = 1 , . . . , s . Assume the contrary and let A 0 be a comp onent of ∂ P \ Σ + ( G 0 ) which a ttains the inﬁmum in the deﬁnitio n of s z ( G 0 ). By Lemma 7.10, ∂ A 0 is a simple closed curve con tained in one of the comp onents C m . Assume for instance that ∂ A 0 ⊂ C 1 and let i ∈ { 1 , . . . , s } sa tisfy (7.12) y 1 6 = y i . W e prov e that there exists a p ositive or neg ative face G 1 of P s uch that (7.13) sz ( G 1 ) < sz ( G 0 ) . After po ssibly substituting P ′ with P ′ G 0 , we may assume G 0 po sitive and x + ( G 0 ) = 0. Let us deﬁne G 1 . Choose an edge S 1 of C i and let F 1 be the facet of P asso ciated to S 1 by Lemma 7.8. When S 1 is an edge o f P and P ′ we deﬁne G 1 = S 1 , otherwise we deﬁne G 1 = F 1 . By L e mma 7 .8, G 1 is nega tive and x − ( G 1 ) = y i . Ther e fore P G 1 = − P ′ + y i . Mor eov er , by deﬁnition, F ( G 1 ) = THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 27 { edges or facets F of P which are neutral or nega tive a nd satisfy x − ( F ) 6 = y i } . Let us prove (7.14) Σ + ( G 1 ) ∩ A 0 = ∅ . Let S be an edge of ∂ A 0 and let F b e the facet of P a s so ciated to S by Lemma 7.8. By Lemma 7.8, either S is a n edge of P and P ′ , is neg ative a nd x − ( S ) = y 1 , or else F is neg ative and x − ( F ) = y 1 . Since y 1 6 = y i , in the ﬁrst case we have S ∈ F ( G 1 ), while in the se c o nd case w e hav e F ∈ F ( G 1 ). In bo th cases we ha ve S ∩ Σ( G 1 ) = ∅ . Thu s we have ∂ A 0 ∩ Σ( G 1 ) = ∅ . Since Σ + ( G 1 ) is the c losure o f a comp onent of Σ( G 1 ), the previo us identit y implies either Σ + ( G 1 ) ⊂ cl A 0 or (7.14). Since Σ + ( G 1 ) contains relint G 1 (b y Lemma 7.6), and this set meets C i , which do es not intersect cl A 0 , we have (7.1 4). Let A 1 be the comp onent of ∂ P \ Σ( G 1 ) which contains A 0 . In order to prov e (7.13) it suﬃces to prove size ( ∂ P \ A 1 ) < size ( ∂ P \ A 0 ). Ass ume the contrary , that is, in view of the inclusion A 0 ⊂ A 1 , a ssume size ( ∂ P \ A 1 ) = size ( ∂ P \ A 0 ) . Let S and F b e as above. The previous e q uality implies (7.15) (in t F ) ∩ ∂ P \ A 1 6 = ∅ , since (int F ) ∩ ∂ P \ A 0 ⊃ (in t F ) ∩ Σ + ( G 0 ) 6 = ∅ . Let us pr ov e the following claims: (i) b oth S a nd − S + y 1 are edges of P , P ′ , − P + y 1 and − P ′ + y 1 , condition (7.4) (with y = y 1 ) ho lds and w e hav e ∂ A 0 ∪ ( − ∂ A 0 + y 1 ) ⊂ ∂ P ∩ ∂ P ′ ; (ii) the facet F contains a trans la te of y i − y 1 ; (iii) at each po int o f ∂ A 0 ∪ ( − ∂ A 0 + y 1 ) a line parallel to y i − y 1 suppo rts P and P ′ ; (iv) there exists a neighbourho o d of relint S such that in that neighbourho o d we ha ve ∂ P = ∂ P ′ on o ne side o f S and ∂ P ∩ ∂ P ′ = ∅ on the other side of S . The same prop erty ho lds for some neighbourho o d o f r elint ( − S + y 1 ). T o prov e (i ), let us show t hat S is an edge of P a nd of P ′ . If this is no t true then, as prov ed ab ov e, we hav e F ∈ F ( G 1 ) and, therefore, F ∩ Σ( G 1 ) = ∅ . This implies (in t F ) ∩ Σ + ( G 1 ) = ∅ . Since int F is clea rly path-connected to A 0 (through S ), we hav e in t F ⊂ A 1 , which contradicts (7.1 5). The rest of (i) follows by Lemma 7.8. T o prove (ii) o bserve that, by (7.4), − P ′ + y 1 has a facet F ′′ coplanar to F . If p ∈ (int F ) ∩ ∂ P \ A 1 , then the segment connecting p to a p oint q of S has to meet ∂ A 1 , b ecause q is an accumulation p oint of A 0 ⊂ A 1 . A fortior i F contains a segment S ′ of ∂ Σ + ( G 1 ). By Le mma 7.8, applied to S ′ and to the pa ir P and P ′ G 1 , ther e is a facet of P ′ G 1 = − P ′ + y i coplanar to F . This facet necessa rily coincides with F ′′ + y i − y 1 . This implies the claim. Claim (iii) follows by (ii) and the ex istence of a facet of P ′ coplanar to F , which is prov e d in Lemma 7.8. Claim (iv) follows by (7.4), with y = y 1 . Let ∆ b e the cylindrical surface which suppo r ts P and with ge ner atrix paralle l to y i − y 1 and let l deno te a g eneric line contained in ∆. W e hav e F ⊂ ∆ a nd ∂ A 0 ∪ ( − ∂ A 0 + y 1 ) ⊂ ∆, by (ii) and (iii). Moreover, Claim (iii ) implies (7.16) [ p, q ] ⊂ ∂ P ∩ ∂ P ′ whenever p, q ∈ l ∩  ∂ A 0 ∪ ( − ∂ A 0 + y 1 )  . Let f b e an homeomo rphism b etw een S 1 and ∂ A 0 and let h = g ◦ f , where g : ∆ → ∆ | ( y 1 − y i ) ⊥ is the orthogonal pr o jection. The set ∆ | ( y 1 − y i ) ⊥ is homeomorphic to S 1 and h can b e seen as a map from S 1 to S 1 . Its winding nu mber is 0, 1 or − 1, since the curve ∂ A 0 is simple. Assume that the winding num be r of h is 0. In this case h is homotopic to a constant map. Since the pro jection g is the identit y b etw een the ﬁrst homotopy groups o f ∆ and of ∆ | ( y 1 − y i ) ⊥ , a lso f , as a ma p from S 1 to ∆, is homotopic to 28 GABRIELE BIANCHI the co nstant map. In this case ∆ \ ∂ A 0 contains a b ounded co mpo nent N . Let us prov e (7.17) Σ + ( G 0 ) = cl N . The pr op erty (7.16) implies N ⊂ ∂ P ∩ ∂ P ′ . If l is as ab ov e and l intersects transversally the rela tive interior o f an edge of ∂ A 0 , then l ∩ N is co nt ained in the facet of P and in the facet o f P ′ asso ciated to this edge. Since their in tersection is contained in Σ + , we have l ∩ N ⊂ Σ + . A contin uity arg ument implies cl N ⊂ Σ + . T o prov e the converse inclusion, observ e that ∂ A 0 ∩ Σ( G 0 ) = ∅ , b ecause no point in ∂ A 0 has a neighbourho o d co nt ained in ∂ P ∩ ∂ P ′ , b y (iv). There fore Σ + ( G 0 ) is contained in c l N or in ∂ P \ N . The inclusio n cl N ⊂ Σ + ( G 0 ) implies cl N = Σ + ( G 0 ). The previous argument also sho ws tha t Σ + ( G 0 ) is conv ex in the y 1 − y i direction. This pro pe r ty and the co nnectedness of Σ + ( G 0 ) imply tha t ∂ Σ + ( G 0 ) has o nly one comp onent. This contradicts (7.1 2). Now assume that the winding num b er of h is 1 or − 1. Cho o se a n or ient ation on S 1 and on ∆ | ( y 1 − y i ) ⊥ . W e claim that h is non-increasing or non-decreasing. If not, there is a line l in ∆ which in tersects tra nsversally the relative interior of a t least three segments S 1 , S 2 and S 3 of ∂ A 0 . These segments are necessa rily contained in the same facet o f ∆, and, if the p oint S 2 ∩ l lies in b etw e en S 1 ∩ l and S 3 ∩ l , then S 2 ∩ l is in the interior o f the q ua drilatera l conv ( S 1 ∪ S 2 ), which is contained in ∂ P ∩ ∂ P ′ . This contradicts (iv). Let us prov e that h is s tr ictly increasing or strictly decreasing. Assume the co ntrary . Then there exist consecutive segments S 1 , S 2 and S 3 of ∂ A 0 , with S 2 parallel to the genera trix of ∆ and S 1 and S 3 on opp osite sides of the line containing S 2 . Since b o th triangles conv ( S 1 ∪ S 2 ) and conv ( S 2 ∪ S 3 ) a re c ontained ∂ P ∩ ∂ P ′ (b y Claims (i) and (iii)), P and P ′ coincide on b oth sides of S 2 . This contradicts Claim (iv). Thu s ea ch line l ⊂ ∆ intersects ∂ A 0 (and also − ∂ A 0 + y 1 ) in exactly o ne point. Arguing a s in the pro of of (7.1 7) one shows that Σ + ( G 0 ) is th e union of the segments parallel to y 1 − y i with endpo int s in ∂ A 0 ∪ ( − ∂ A 0 + y 1 ). Thus ∂ Σ + ( G 0 ) has at most t wo comp onents C 1 = ∂ A 0 and C 2 = − ∂ A 0 + y 1 and w e hav e i = 2. Each edg e of − ∂ A 0 + y 1 = C 2 is an edg e of P and P ′ , by Claim (i), and this implies that G 1 is an edge of P and P ′ . Claim (i) also implies that G 1 is an edge of − P ′ + y 1 and cone ( P , G 1 ) = cone ( − P ′ + y 1 , G 1 ) . Lemma 7.8 prov es that G 1 is als o an edge of − P ′ + y 2 and that cone ( P , G 1 ) = cone ( − P ′ + y 2 , G 1 ) , since G 1 is an edge of C 2 . The previous tw o identities imply y 1 = y 2 , contradicting (7.12).  Lemma 7. 13. Assume that P is n ot a tr anslation or a r eﬂe ction of P ′ . L et G 0 b e a p ositive or ne gative fac e of P s uch that the ve ctors asso ciate d by L emma 7.9 to e ach c omp onent of ∂ Σ + c oincide, and let y denote this ve ctor. Then ther e exists a c omp onent of ∂ P \ ( − ∂ Σ + + y ) interse cting − Σ + + y whose b oun dary is − ∂ Σ + + y . If Σ − denotes the closur e of this c omp onent t hen we have (7.18) Σ − ⊂ ∂ P ∩ ∂ P ′ G 0 and Σ − 6 = − Σ + + y . Remark 7.14. W e recall tha t Σ + ⊂ ∂ P ∩ ∂ P ′ G 0 , by deﬁnition, and that P , P ′ , − ( P G 0 ) + y a nd − ( P ′ G 0 ) + y co incide in an o ne - sided neighbourho o d of ∂ Σ − = − ∂ Σ + + y , by Lemma 7.9. Pr o of. Up to a tra nslation or a reﬂection o f P ′ , we may a s sume G 0 po sitive and P ′ G 0 = P ′ . Cho ose p 0 ∈ relint P ∩ relint ( − P + y ) and consider the homeomor phism from − ∂ P + y to ∂ P which a sso ciates to p ∈ − ∂ P + y the intersection o f ∂ P with THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 29 the ray issuing from p 0 and con ta ining p . T his homeomor phism maps int ( − Σ + + y ) in a connected subset of ∂ P intersecting − Σ + + y and whose b oundary is − ∂ Σ + + y . Similar arguments prov e tha t there ex ists a comp onent (whose c lo sure w e denote by Σ ′ − ) o f ∂ P ′ \ ( − ∂ Σ + + y ) intersecting − Σ + + y whose b o undary is − ∂ Σ + + y . Let us show that the se t V P of the vertices of P in Σ − coincides with the set V P ′ of the vertices of P ′ in Σ ′ − . Let w ∈ S 2 be such that P w ∈ V P . Up to a per turbation o f w , we may as s ume that ( − P + y ) w is a vertex to o. Since P and − P + y coincide in an o ne-sided neighbour ho o d of ∂ Σ − = − ∂ Σ + + y a nd they are conv ex, ( − P + y ) w is cont ained in − Σ + + y . Since − Σ + ⊂ ∂ ( − P ) ∩ ∂ ( − P ′ ) w e hav e ( − P ) w = ( − P ′ ) w . The latter, the identit y D P = D P ′ and (7.1) imply P w = P ′ w and prove V P ⊂ V P ′ . Similar arguments pr ove V P ⊃ V P ′ . The identit y V P = V P ′ implies Σ − = Σ ′ − and Σ − ⊂ ∂ P ∩ ∂ P ′ G 0 . Assume Σ + 6 = − Σ − + y false. Since G 0 ⊂ Σ + = − Σ − + y , G 0 is als o a face of P ′ and of − P ′ + y . Moreov er, the inclusion r elint G 0 ⊂ int Σ + , pr ov ed in Le mma 7.6, implies cone ( P , G 0 ) = cone ( P ′ , G 0 ) = cone ( − P ′ + y , G 0 ). This pr ov es that G 0 is neutral, co ntradicting the deﬁnition of G 0 .  8. Proof of the main theorem Pr o of of The or em 1.1. W e assume that P ′ is not a translation or a reﬂection o f P . The pair s ( P , − P ) and ( P ′ , − P ′ ) ar e sy nisothetic, by Theorem 1.2. Lemma 7.12 applies a nd proves that there e x ists a p ositive or negative fa ce G 0 of P s uch that the vectors as so ciated by Lemma 7 .9 to each comp onent of ∂ Σ + coincide. Let y denote this vector and let Σ − be deﬁned as in Lemma 7 .13. Up to a translatio n or a r eﬂection of P ′ , we may assume G 0 po sitive and P ′ G 0 = P ′ . W e will prov e g P 6 = g P ′ . In this proof the terms boundary and int erior, when applied to Σ + and Σ − , refer to the top ology induced on ∂ P by its immers io n in R 3 . W e cla im that either there exist q ∈ int ( − Σ − + y ) vertex of − P + y and a pla ne π strictly s upp or ting − P + y at q such that π ∩ P = ∅ , or else ther e exist q ∈ int Σ + vertex of P and a plane π strictly suppor ting P at q such tha t π ∩ ( − P + y ) = ∅ . This follows by (7.18) and standar d conv ex it y ar g ument s. Indeed, the set V P of the vertices of P contained in int Σ + diﬀers fro m the set V − P + y of the vertices of − P + y co nt ained in int ( − Σ − + y ), b ecause otherwise b oth Σ + and − Σ − + y are contained in the b o unda ry of conv ( ∂ Σ + ∪ V P ) and the inequalit y in (7.18) is false. If, s ay , q ∈ V − P + y \ V P then le t π b e a plane through q strictly suppo rting − P + y . Up to a p ertur ba tion of π , we may assume that either π do es not in tersect P or it intersects int P . In the ﬁrs t case we a re done. In the second cas e there is an “extremal” vertex of P contained in the op en halfspa ce which is b ounded by π and do es not contain − P + y . This vertex has the req uired pro p e rties. Assume q ∈ int ( − Σ − + y ) (in the other case the pro of is similar) a nd let π be as ab ov e. Le t π 0 be a plane which is parallel to π , intersects ∂ Σ + in a p oint, say z (up to a pertur ba tion o f π we may always a ssume tha t π 0 ∩ Σ + is a p oint), and suc h that ∂ Σ + and q are o n opp osite sides of π 0 . If π + 0 denotes the closed halfplane b ounded by π 0 and co ntaining q , then we ha ve π + 0 ∩ ∂ P ⊂ Σ + and π + 0 ∩ ( − ∂ P + y ) ⊂ Σ − . Therefore, by the inclusion in (7 .1 8), we hav e (8.1) π + 0 ∩ P = π + 0 ∩ P ′ and π + 0 ∩ ( − P + y ) = π + 0 ∩ ( − P ′ + y ) . The plane − π + y strictly supp orts P and P ′ at − q + y . Up to aﬃne transfo r - mations, we may assume − q + y = O , − π + y = { x ∈ R 3 : x 3 = 0 } , P, P ′ ⊂ { x : 0 ≤ x 3 ≤ 1 } , and we may also assume that the plane { x : x 3 = 1 } supp orts bo th P and P ′ (the existence of a common suppo rting plane follows by (8.1)). In this setting we hav e y · (0 , 0 , 1) > 1, b ecaus e y = q a nd the plane π , which contains q , 30 GABRIELE BIANCHI do es no t intersect P and P ′ . Let e 3 = (0 , 0 , 1 ) and let N 1 and N 2 denote the strips { x : z · e 3 ≤ x 3 ≤ 1 } and { x : 0 ≤ x 3 ≤ ( − z + y ) · e 3 } , resp ectively . The identities (8.1) a r e equiv a lent to (8.2) P ∩ N 1 = P ′ ∩ N 1 and P ∩ N 2 = P ′ ∩ N 2 . W e cla im tha t g P and g P ′ diﬀer in any neighbourho o d o f z . Consider P ∩ ( P + z + ε ) and P ′ ∩ ( P ′ + z + ε ), for ε ∈ R 3 , k ε k s mall and ε · e 3 < 0. These sets are contained in the strip N 1 ∪ N ( ε ), where N ( ε ) = { x : ( z + ε ) · e 3 ≤ x 3 ≤ z · e 3 } . Since N 1 ⊂ N 2 + z + ε (beca use ε · e 3 < 0 and y · e 3 > 1 ), (8 .2) implies P = P ′ and P + z + ε = P ′ + z + ε in N 1 . In N ( ε ) we have P + z + ε = P ′ + z + ε , but P and P ′ diﬀer. T o b e more precise, let A = cone ( P , O ) = cone ( P ′ , O ), C = cone ( P, z ) ∩ { x : x 3 ≤ 0 } and D = cone ( P ′ , z ) ∩ { x : x 3 ≤ 0 } . W e hav e ( P + z + ε ) ∩ N ( ε ) = ( P ′ + z + ε ) ∩ N ( ε ) = ( A + z + ε ) ∩ N ( ε ) . W e also hav e P ∩ ( P + z + ε ) ∩ N ( ε ) = ( A + z + ε ) ∩ ( C + z ) , P ′ ∩ ( P ′ + z + ε ) ∩ N ( ε ) = ( A + z + ε ) ∩ ( D + z ) , bec ause O is a vertex of A and therefore the in tersections in the left hand side of the pr evious formulas a r e contained in a neighbour ho o d of z . Therefore g P ( z + ε ) − g P ′ ( z + ε ) = g A,C ( ε ) − g A,D ( ε ) . If we prove that C 6 = D and that O is a vertex of conv ( C ∪ D ), then Lemma 3 .6 (with B = { O } ) implies g P 6 = g P ′ . Let us prove (8.3) cone ( P, z ) 6 = cone ( P ′ , z ) . By deﬁnition, z is an e ndp oint of an edge S of ∂ Σ + which intersects { x : z · e 3 = x 3 } only in z . Let F and F ′ be resp ectively the facets of P and P ′ asso ciated to S by Lemma 7.8. If (8.3) is fals e then either z is a vertex of P a nd P ′ , or z is in the int erior of F and F ′ , or z is in the relative interior of tw o edges of P and P ′ . The last t wo possibilitie.s are ruled out b y Lemma 7.8, whic h prov e s that z is a v ertex of the po lygon F ∩ F ′ . The ﬁrst p o s sibility is ruled out by arguments similar to those used in the pro of of Lemma 7.6. These a r guments prove that when z is a vertex the ident ity co ne ( P, z ) = co ne ( P ′ , z ) is not compatible with z ∈ ∂ Σ + . F o rmula (8.3) and the equa lity cone ( P, z ) ∩ { x : x 3 ≥ 0 } = cone ( P ′ , z ) ∩ { x : x 3 ≥ 0 } (a consequence of (8.2)) imply C 6 = D . It remains to prov e that O is a vertex of conv ( C ∪ D ). T he ﬁrst condition in (8.2) implies that C ∩ { x : x 3 = 0 } = D ∩ { x : x 3 = 0 } . Ther efore O is no t a vertex of conv ( C ∪ D ) only if z is in the relative interior of a s egment R contained in ∂ P , in ∂ P ′ and in { x : z · e 3 = x 3 } . Let S , F and F ′ be as ab ov e. Since S ⊂ ∂ P ∩ ∂ P ′ and S * { x : z · e 3 = x 3 } , T := conv ( R ∪ S ) is a triangle contained in ∂ P ∩ ∂ P ′ . The latter contradicts th e prop erty that S is an edge of the po lygon F ∩ F ′ , becaus e relint S ⊂ relint T ⊂ F ∩ F ′ .  9. About dimension n ≥ 4 Each conv ex bo dy has a representation as in (2.1) in terms of directly indecom- po sable b o dies K i . Assume that a t least tw o of the s umma nds , say K 1 and K 2 , are not centrally symmetric. In this cas e ( − K 1 ) ⊕ K 2 ⊕ · · · ⊕ K s has the sa me cov ar iogra m of K and is not a tr anslation o r r eﬂection of K . Do es ea ch conv e x bo dy L with g L = g K hav e the sa me structure ? Theorem 9. 1. L et K ⊂ R n b e a c onvex b o dy and let K = K 1 ⊕ · · · ⊕ K s b e its r epr esentation in t erms of dir e ct ly inde c omp osable b o dies. Assume that, for e ach i = 1 , . . . , s , E i is a line ar subsp ac e c ontaining K i and E 1 ⊕ · · · ⊕ E s = R n . THE CO V ARIOGRAM DE TERMINES CONVEX 3-POL YTOPES 31 (i) If L is a c onvex b o dy and g K = g L then L = L 1 ⊕ · · · ⊕ L s , wher e, for e ach i , L i is a dir e ctly inde c omp osable c onvex b o dy c ont aine d in E i and g K i = g L i . (ii) If in addition K is a p olytop e and, for e ach i , either dim K i ≤ 3 , or K i is c entr al ly symmetric, or K i is simplicial with K i and − K i in gener al r elative p osition, then L is a tr anslation or r eﬂe ct ion of σ 1 K 1 ⊕ · · · ⊕ σ s K s , for suitable σ 1 , . . . , σ s ∈ { +1 , − 1 } . Lemma 9.2. A c onvex b o dy K is dir e ctly inde c omp osable if and only if D K is dir e ctly inde c omp osable. Pr o of. It is evident that if K is not directly indecompo sable then D K has this prop erty to o . Vice versa, assume that D K = L ⊕ M , with L ⊂ E 1 , M ⊂ E 2 conv ex sets of s trictly p ositive dimension, and E 1 , E 2 linear subspa c es with E 1 ⊕ E 2 = R n . Up to a linear tra nsformation we may assume E 1 = E ⊥ 2 . In this case we hav e h DK ( x 1 , x 2 ) = h DK ( x 1 , 0) + h DK (0 , x 2 ), for each x 1 ∈ E 1 and x 2 ∈ E 2 . Moreov er, if K 1 = K | E 1 and K 2 = K | E 2 , we a lso have h K 1 ( x 1 , 0) = h K ( x 1 , 0) and h K 2 (0 , x 2 ) = h K (0 , x 2 ). The linearity of the s upp or t function with resp ect to Minko wski a ddition implies h K 1 ( x 1 , 0) + h − K 1 ( x 1 , 0) = h K ( x 1 , 0) + h − K ( x 1 , 0) = h DK ( x 1 , 0) and h K 2 (0 , x 2 ) + h − K 2 (0 , x 2 ) = h DK (0 , x 2 ). Thes e equalities imply h K 1 ⊕ K 2 ( x 1 , x 2 ) + h − ( K 1 ⊕ K 2 ) ( x 1 , x 2 ) = h DK ( x 1 , x 2 ) = h K ( x 1 , x 2 ) + h − K ( x 1 , x 2 ) , which can b e rewritten a s (9.1) h K 1 ⊕ K 2 ( x 1 , x 2 ) + h K 1 ⊕ K 2 ( − x 1 , − x 2 ) = h K ( x 1 , x 2 ) + h K ( − x 1 , − x 2 ) . The inclusion K ⊂ K 1 ⊕ K 2 , whic h is obvious, implies h K ≤ h K 1 ⊕ K 2 , and the la tter inequality , together w ith (9.1), implies h K = h K 1 ⊕ K 2 . Therefor e K = K 1 ⊕ K 2 is not indecompo sable to o.  Pr o of of The or em 9.1. Cla im (i). The identit y D K = D L , Lemma 9.2 and the uniqueness o f the decomp ositio n in directly indecomp osa ble summands imply L = L 1 ⊕ · · · ⊕ L s , w ith L i ⊂ E i . The identit y Π s i =1 g K i ( x i ) = g K ( x 1 , . . . , x s ) = g L ( x 1 , . . . , x s ) = Π s i =1 g L i ( x i ) , v alid for each x = ( x 1 , . . . , x s ) ∈ E 1 ⊕ · · · ⊕ E s , implies g K i = α i g L i , for each index i and for suitable co ns tants α i , α i 6 = 0. Lemma 4.3 implies α i = 1, for each i . Claim (ii) follows from Claim (i), Theorem 1.1 and the p ositive r esults for the cov ar iogra m problem av a ilable in the liter ature and mentioned in the introduction.  Remark 9.3. Let P and P ′ be convex p olyto pes in R 4 with non-empty interior and with equal cov ar iogra m, le t w ∈ S 3 and assume that P w and P − w are facets. Prop ositio n 4.1, Lemma 4.3 and Theorem 1.1 imply that, po ssibly after a reﬂection or a tr anslation of P ′ , we have P ′ w = ± P w and P ′ − w = ± P − w . Contrary to the three-dimensional case (see also Remar k 4.4), the ambiguity due to the ± sign cannot b e elimina ted. Indeed, if K = con v { (0 , − 2) , (0 , 2) , (1 , 1) , (1 , − 1) } , L is a triangle, P = K × L , P ′ = K × ( − L ) and w = ( − 1 , 0 , 0 , 0 ) then P ′ w = − P w but there is no translation or reﬂection of P ′ such that P ′ w = P w . 32 GABRIELE BIANCHI References AP91. R. J. A dler and R. Pyk e, Pr oblem 91–3 , Inst. Math. Statist. 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