Construction and Analysis of Projected Deformed Products

Construction and Analysis of Pro jected Deformed Pro ducts Raman San y al G ¨ un ter M . Ziegler Institute of Mathematics, MA 6-2 TU Ber lin D-10623 Berlin, Germa ny { sany al,zie gler } @math.tu-berlin.de Octob er 10, 2007 Abstract W e introduce a deformed pr o duct construction for simple p olytop es in terms of lower- triangular blo ck matrix represen tations. W e further show how Gale duality can b e employ ed for the constr uction and for the a nalysis of deformed pro ducts such that sp ecified faces (e.g. all the k -faces) are “ strictly pres erved” under pro jection. Thu s, starting from an ar bitrary neighborly simplicial ( d − 2 )-po lytop e Q on n − 1 vertices we construct a deformed n -cub e, whose pr o jection to the last d co or dinates yields a neighb orl y cubic al d -p olytop e . As an extension o f the cubical ca se, we constr uct matrix r epresentations of deformed products of (ev en) polyg ons (D PPs), which hav e a pro jection to d -spa ce that retains the complete ( ⌊ d 2 ⌋ − 1)-skeleton. In b oth case s the combinatorial structure of the images under pro jection is completely determined by the neig h b orly p olytop e Q : Our analysis provides explicit combinatorial de- scriptions. This yields a multitude of combinatorially different neighbo r ly cubical p olytop es and DPPs. As a sp ecial case, we obtain simplified descr iptions of the neighborly cubical p olytop es of Joswig & Ziegler (2000) a s well a s o f the pr oje cte d deforme d pr o ducts of p olygo ns that were announced b y Ziegler (20 04), a family of 4-p olytop es who se “ fatness” gets arbitrar ily close to 9. 1 In tro duction Some r emark able geometric effects can b e ac hiev ed f or p ro jections of “suitably-deformed” high- dimensional simple p olytop es. This includes the Klee-Min t y cub es [7], the Goldfarb cub es [3], and man y other exp onen tial examples for v arian ts of the simplex algorithm, but also the “neigh- b orly cu bical p olytop es” first constructed by Joswig & Ziegler [6]. A geometric framew ork for “deformed p ro duct” constructions was provided b y Amenta & Ziegler [1]. Here w e in tro du ce a gene r alize d deforme d pr o ducts constru ction. In terms of this constru ction, the previous v ersion b y Ament a & Zieg ler conce rned deformed prod ucts of rank 1. The new construction is presented in matrix v ersion (that is, as an H -p olyto p e). Iterated deformed pro du cts are th us giv en b y lo w er-triangular blo c k matrices, where the blo cks b elo w the diagonal do not infl u ence the com binatorics of the pro duct (for suitable r igh t-hand s id es). 1 The deformed pro d ucts P are constructed in order to pro vide int eresting images after an affine pro jection π : P → π ( P ). The d eformations we are after are designed so that certain cla sses of fac es of th e d eformed pro duct P , e.g. all the k -faces, are “preserv ed” b y a pro jection to some lo w-dimensional space, i.e. m app ed to faces of π ( P ). In the co m binatorially-con v enien t situation, the faces in question are strictly pr eserve d b y the pro jection; we giv e a linear algebra condition that c haracterizes th e faces that are strictly pr eserv ed (Pro jection Lemm a 2.5). W e also identify a situation w hen all nontrivial faces of π ( P ) arise as images π ( F ) of faces F ⊂ P that are strictly preserved (Corollary 2.8). The conditions dictated b y the Pro jection Lemm a ma y b e translated v ia a non-standard app li- cation of Gale du alit y [4 , Sect. 6.3] [15 , Lect. 6] in to conditions ab out the combinatorics of an auxiliary p olytop e Q . As an instance of this set-up, w e sho w ho w neighb o rly cubic al d -p olytop es a rise from pr o jections of a deformed n -cub e w here all the ( ⌊ d 2 ⌋ − 1)-faces are preserved by th e pro jection. Th e p recise form of the matrix representati on of the n -cub e, and th e com binatorics of th e r esulting p olytopes, is dictated via Gale dualit y b y a n eigh b orly simp licial (!) ( d − 2)- p olytop e with n − 1 v ertices. As sp ecial cases, we obtain the neigh b orly cubical p olytop es firs t obtained by Joswig & Ziegler [6], and also geometric realizations for neigh b orly cubical spheres as d escrib ed by Joswig & R¨ orig [5]. Finally , we construct and analyze pr oje cte d deforme d pr o ducts of (even) p olygons (PDPP p oly- top es), as th e images of a d eformed pro du ct of r even p olygons, pr o jected to R d . The pro jection is d esigned to strictly preserv e all the ( ⌊ d 2 ⌋ − 1)-fa ces (as w ell as add itional d 2 -faces, if d is ev en ). This pro duces in p articular the 2-parameter family of 4-dimen s ional p olytop es from [17], for whic h the “fatness” parameter in tro d uced in [16] gets as large as 9 − ε . W e p resen t a new construction (dr astically simplified and systematized) and a complete com bin atorial description of these p olytop es. This work is b ased on the Diploma T hesis [12]; see also the researc h announ cemen ts in [17] and [18]. The “w edge pro d uct” p olyto p es of R¨ orig & Z iegler [11] p ro vide another in teresting instance of “deformed high-dimen sional simple p olytop es”. A f urther analysis shows that the n eigh b orly cubical p olytop es, the PDPP p olytop es as w ell as th e we dge pro ducts do exhibit a wealth of in teresting p olyhedral s u rfaces, including the “surfaces of u nusually high gen us” by McMullen, Sc h ulz & Wills [9], and equivel ar su rfaces of t yp e ( p, 2 q ). T op ological obstru ctions that preven t a suitable pr o jectio n of “deformed p ro ducts of o dd p olygons”, or of the wedge pro du ct p olytop es, will b e presente d b y R¨ orig & Sany al [10]. Ac knowledgemen ts. The fi rst author wo uld lik e to th ank And reas P affenholz, T hilo R¨ orig, Jak ob Uszko reit, Arn old W aßmer, and Axel W erner for “activ ely listening” and V anessa K¨ a¨ ab for more. Both authors gratefully ac knowle dge supp ort by th e German Science F oundation DF G via the Researc h T raining Group “Metho ds for Discrete Stru ctur es” and a Leibniz gran t. 2 Basics In this section we reca ll basic prop erties and nota tion ab out the main ob j ects of this pap er: c onvex p olytop es . Readers new to the country of p olytopia will fin d usefu l inform ation in the w ell-kno wn tra vel guides [4] and [15] while the frequent visitors might wish to skim the section for p ossibly non-standard notation. One of the main messages this article tries to conv ey is th at it p a ys off to work with p olyto p es in explicit c o or dinates (matrix represent ation). Classically , there a re t wo fundamen tal wa ys of viewing a p olytop e in co ordinates: the in terior or V -rep r esen tation, and the exterior or H -repr esentati on. F or V -p olytop es “with few v ertices”, Perles [4, Chap. 6] had devel op ed Gale 2 dualit y as a p ow erful to ol. In this article, w e w ill apply Gale dualit y for th e analysis of pro jected simple H -p olytop es. Th e basics for this will b e dev elop ed in this s ection. 2.1 P olytop es in co ordinates F or the rest of the section, let P ⊂ R d b e a full-dimens ional polytop e. In its interior or V - pr esentation , P = conv V is giv en as the conv ex hull of a finite p oin t set V = { v 1 , . . . , v m } ⊂ R d and V is inclusion-minimal with resp ect to th is prop erty . The elemen ts of V are called vertic es , with notation vert P = V . F or a non emp t y subset I ⊆ [ m ] = { 1 , . . . , m } th e set V I = { v i : i ∈ I } forms a fac e of P if there is a linear functional ℓ : R d → R su c h that ℓ attains its maxim um o ver P on F = conv V ′ . The d imension dim F is the dimension of its affine span. The emp t y set is also a face of P , of dimens ion − 1. The collection F P of all faces of P , ordered b y inclusion, is a graded, atomic and coatomic lattice w ith dim + 1 as its rank function. W e denote b y F ∂ P := F P \ { P } the face p oset of the b ou n dary of P . W e sa y that tw o p olytopes are of the same c ombinatoria l typ e if their face lattices are isomorph ic as abstract p osets. A p olytop e P is simplicial if small p erturb ations ap p lied to the ve rtices do n ot alter its com binatorial t yp e. Equiv alen tly , ev er y k -face of P ( k < di m P ) is the con v ex hull of exactly k + 1 vertic es. The quotient P /F of P by a f ace F is a p olytop e with face lattice isomorphic to F P ≥ F = { G ∈ F P : F ⊆ G } . If F = { v } is a ve rtex, then P /v is called a vertex figur e at v . The p olytope P is give n in its exterior or H - pr esentation if P is the inte rsection of fi nitely many halfspaces. T hat is, if there are (outer) normals a 1 , . . . , a n ∈ R d and displacemen ts b 1 , . . . , b n ∈ R suc h that P = n \ i =1 { x ∈ R d : a T i x ≤ b i } , where w e assum e that the collectio n of norm als is irr e dundant , thus discarding any on e of the halfspaces c hanges the p olyto p e. Th e hyp erplanes H i = { x ∈ R d : a T i x = b i } are said to b e fac et defining ; the corresp onding ( d − 1)-faces F i = P ∩ H i are called fac ets . More compactly , we think of the normals a i as th e r o ws of a m atrix A ∈ R n × d and, w ith b ∈ R n accordingly , write P = P ( A, b ) = { x ∈ R d : A x ≤ b } . F or an y s ubset F ⊆ P let eq F = { i ∈ [ n ] : F ⊂ H i } ⊆ [ n ] b e its e quality set . Cle arly , F ⊆ T i ∈ eq F F i ; in case of equ ality , the set F is a face of P . De note by A I the su bmatrix of A induced by the r ow in dices in I ⊆ [ n ]. Thus any face F is giv en by F = P ∩ { x : A I x = b I } , for I = eq F . The collection of equalit y sets of faces ord ered by rev erse inclusion is isomorphic to F P . The p olytope P is simple if its com binatorial type is stable und er small p erturbations applied to the b ounding hyp erplanes. Equiv alently , ev ery nonemp t y face F is con tained in no more than | eq F | = d − di m F facets. 2.2 Gale duality Let P ⊂ R d b e a d -p olytop e an d let the rows of V ∈ R m × d b e the m vertic es of P . Denote by V hog = ( V , 1 ) ∈ R m × ( d +1) the homogenizati on of V . The column span of V is a d + 1 dimensional linear su bspace. Cho ose G ∈ R m × ( m − d − 1) suc h that the columns form a basis for the orthogonal complemen t. Any su c h basis, regarded as an ordered collectio n of m ro w vec tors, is called a Gale tr ansform of P . I t is unique up to linear isomorph ism and, by the r ev erse pro cess, c haracterizes V hog , again u p to linear isomorphism. So it determines P only up to a pro jectiv e transf ormation. Ho wev er, the striking feature of Gale transf orms is that its com b inatorial prop erties are, in a precise sense, du al to those of P ; this corresp ond ence go es by th e n ame of Gale duality . 3 In order to state and w ork with Gale dualit y w e in tro d uce some concepts and notations. As b efore we w r ite V I for the su bset of the r o ws of V indexed by I ⊆ [ m ]. A su bset I ⊂ [ m ] names a c ofac e of P if the complement V [ m ] \ I is the verte x set of a f ace of P . Definition 2.1. A collection of v ectors G = { g 1 , g 2 , . . . , g m } ⊂ R k is p ositively dep endent if there are num b ers λ 1 , λ 2 , . . . , λ m > 0 s uc h that λ 1 g 1 + · · · + λ m g m = 0. It is p ositively sp anning if in addition G is of full rank k . “Begin p ositiv ely span n ing” is, like “b eing spannin g”, an op e n condition, i.e. preserved under (sufficien tly small) p erturbations of the elemen ts of G . Th is, ho w ev er, is not tru e for “b eing p ositiv ely dep endent”: Consider e.g. { g , − g } for g ∈ R k , g 6 = 0, k > 1. Theorem 2.2 (Gale du alit y) . L et P = conv V b e a p olytop e and G a Gale tr ansform of P . L et I ⊂ [ m ] then I names a c ofac e of P i f and only if G I is p ositively dep endent. In light of Gale d ualit y , th e p receding theo rem implies that for a general p olytop e not ev ery subset of th e vertex set of a face necessarily forms a face. This, ho w ev er, is true for sim p licial p olytop es and, in fact, c haracterizes th em. A still s tronger condition is satisfied if no d + 1 v ertices of a d -p olytop e lie on a hyp erplane, that is, if the vertic es are in gener al p osition with resp ect to affine h yp erplanes. (Th e p olytope is then automatica lly simplicial; ho w ev er, th e v ertices of a regular o ctahedron are not in general p osition). This translates in to Gale diagrams as follo ws. Prop osition 2.3. L et P ⊂ R d b e a p olytop e and G ⊂ R k a Gale tr ansform of P . Then P is simplicial with ve rtic e s in gener al p osition if and only if the r ows of G ar e in g e ner al p osition with r esp e ct to line ar hyp erplanes, that is, if any k ve ctors of G ar e line arly indep endent. 2.3 F aces strictly preserv ed b y a pro ject ion Pro jections are fundamental in p olytope theory: Ev ery p olyto p e on n v ertices is the image of an ( n − 1)-simplex under an affine pro jection. This in particular sa ys that the analysis of the images of p olytop es under pro jection is as d ifficult as the general classification of all com b in atorial t yp es of p olytopes. T h e p r oblem is th at a k -face F ⊂ P can b eha ve in v arious wa ys und er pro jection: It can map to a k -face, or to p art of a k -face, or to a lo w er-dimensional f ace of π ( P ). Even if it maps to a k -face π ( F ) ⊂ π ( P ), there ma y b e other k -faces of P that m ap to the same face ˜ F = π ( F ). In that case, the face π − 1 ( ˜ F ) has higher dimension than F . Thus, as a serious simplifying measure, w e r estrict our attent ion in the f ollo wing to the most conv enien t situation, of faces that are “strictly p reserv ed” by a pro jection. Definition 2.4 (Strictly pr eserv ed faces [17]) . Let P be a p olyto p e and Q = π ( P ) th e image of P under an affin e pro jection π : P → R d . A nonempty face F of P is (strictly) pr eserve d b y π if (i) π ( F ) is a face of Q combinatoria lly equiv alent to F , and (preserv ed face) (ii) the pr eimage π − 1 ( π ( F )) is F . (strictly preserve d) Since in th e follo w ing w e will b e concerned exc lusive ly with the analysis of strictly p reserv ed faces, we will generally d rop th e mo difi er “strictly” starting no w. The f ollo wing lemma giv es an algebraic w a y to r e ad off th e preserved faces from a p olytope in exterior presentat ion. Ev ery affine pr o jectio n π : R n → R d factors as an affine tr ansformation follo wed a pro jection π d : R n − d × R d → R d that deletes the fi rst n − d co ord in ates, that is π d ( x, x ) = x for all ( x, x ) ∈ R n − d × R d . T h erefore, we will fo cus on the p r o jectio ns π d “to the last d co ord inates”. F or a p olytop e P = P ( A, b ) ⊂ R n in exterior presenta tion the pro jection map π d naturally partitions the columns of A , as A = ( A | A ). 4 Lemma 2.5 (Pro jection Lemma: Matrix v ersion) . L e t P = P ( A, b ) ⊂ R n b e a p olytop e, F a nonempty fac e of P , and I = eq F the index set of the i ne qualities that ar e tig ht at F . Then F is pr eserve d by the pr oje ction π d : P → R d to the last d c o or dinates i f and only if the r ows of A I ar e p ositively sp anning. The p ro of mak es use of th e f ollo wing geometric ve rsion of the F ark as Lemma. Lemma 2.6 ([15, S ect. 1.4]) . L et P = P ( A, b ) b e a p olytop e and F ⊆ P a nonempty fac e. F or a line ar functional ℓ ( x ) = cx we denote by P ℓ the nonempty fac e of P on which ℓ attains its maximum. The line ar function ℓ singles out F , th at is P ℓ = F , if and only if c is a strictly p ositive line ar c ombination of the r ows of A eq F . Pr o of of L emma 2.5. W e sp lit the pro of in to t wo p arts. Claim 1. ˜ F = π d ( F ) is a face of ˜ P = π d ( P ) with π − 1 d ( ˜ F ) ∩ P = F iff A I is p ositiv ely dep end en t. By Lemma 2.6 the ro ws of A I are p ositiv ely dep enden t if and only if there is some c ∈ R d suc h that the linear fu nction ℓ ( x ) := (0 , c ) x = c x s atisfies P ℓ = F . Re writing ℓ = h ◦ π d with h ( x ) = c x we see that suc h a c exists if and only if there is a linear function h on ˜ P such that ˜ P h = ˜ F . Claim 2. Considering F as a (sub-)p olyto p e in its own right, then ˜ F = π d ( F ) is com bin atorially equiv alen t to F if an d only if A I has fu ll r o w rank. The p olytop es F and ˜ F are com binatorially equiv alen t iff they are affinely isomorphic. This happ ens if and only if th e linear map π d is inj ectiv e r estricted the linear space L = { x : A I x = 0 } , whic h is parallel to a ff F = { x : A I x = b I } the a ffine h ull of F . No w, π d | L is injectiv e iff k er π d ∩ L ∼ = { x : A I x = 0 } is trivial. See [13] for a p ro of in a differen t w ording. Lemma 2.5 allo ws us to guarante e that in certain situations every single k -face is preserved by a pro jection π : P → π ( P ). Then, how ev er, we wan t to also see that π ( P ) h as no other k -face than those ind uced by th e p r o jectio n. This will b e argued v ia the follo wing lemma. Lemma 2.7. L et P = P ( A, b ) ⊂ R n b e an n -p olytop e such that for every vertex v ∈ vert P the r ows of the matrix A eq v ar e i n gener al p osition with r esp e ct to line ar hyp erplanes. Then every pr op er f ac e of P is either pr e se rve d under π d or fal ls short of b eing a fac e of π d ( P ) . Pr o of. If G ⊂ R k is a set of at least k vec tors in general p ositio n with r esp ect to linear h yp er- planes then dim span G ′ ≥ min {| G ′ | , k } for eve ry su bset G ′ ⊆ G . In particular, ev ery p ositiv ely dep end ent su bset is p ositiv ely spann ing. Let F ⊂ P b e a prop er face. F rom the pr o of of Lemma 2.5 it follo ws that π d ( F ) is a face iff A eq F is p ositiv ely dep end en t. Let v ∈ vert P b e a ve rtex with v ∈ F . Th en A eq F ⊆ A eq v and A eq v ⊂ R n − d is a set of at least n v ectors in general p ositio n with resp ect to linear hyp erplanes. Corollary 2.8. If all k -fac es of P ar e pr eserve d by the pr oje ction π : P → π ( P ) , then al l k -fac es of π ( P ) arise as images of k -fac e s of P . Pr o of. F or any k -face G ⊆ π ( P ) w e kno w that b G = π d − 1 ( G ) is a face of P , of dimension dim b G ≥ k . No w if F ⊆ b G is any k -face of b G , then b y Lemma 2.7 either F is preserve d, and w e get π d ( F ) = G , or F is n ot mapp ed to a face. T h e latter case cannot arise h er e. 5 2.4 Generalized Deformed Pro ducts The orthogonal pro du ct P × Q ⊂ R d + e of a d -p olytop e P = P ( A, a ) ⊂ R d and an e -p olytop e Q = P ( B , b ) ⊂ R e is giv en in inequalit y descrip tion by a b lo c k d iagonal system: Ax ≤ a B y ≤ b. W e get a deforme d p r o duct (with the combinatoria l s tructure of the orthogonal pro d uct) if w e generalize this into a blo c k low er-triangular system, p ro vided that Q is simple, a nd that w e rescale the right-hand side of the system suitably . Definition 2.9 (Rank r deformed pro duct) . Let P = P ( A, a ) ⊂ R d b e a d -p olytop e and Q = P ( B , b ) ⊂ R e a s imple e -p olytop e, with A ∈ R k × d and B ∈ R n × e . Let C ∈ R n × d b e an arbitrary matrix of r ank r and let M ≫ 0 b e large. The r ank r deforme d pr o duct P ⊲ ⊳ C Q ⊂ R d + e of P and Q with r esp ect to C is giv en by Ax ≤ a C x + B y ≤ M b , that is,  A C B   x y  =  a M b  . Prop osition 2.10. L et P = P ( A, a ) ⊂ R d b e a d -p olytop e, Q = P ( B , b ) ⊂ R e a simple e - p olytop e, P ⊲ ⊳ C Q their deforme d pr o duct, and M > 0 the p ar ameter i nvolve d in its c onstruction. If M is sufficiently lar ge (dep ending on B , b and C ), then P ⊲ ⊳ C Q and P × Q ar e c ombinatorial ly e quivalent. Our pr op osition m ay also b e obtained from the Isomorphism Lemma [1, Lemma 2.4] that wa s applied b y Amenta & Ziegler to p ro v e the co rresp onding statemen t for (r ank 1) deformed pro du cts. Ho w ev er, we us e it in a dual form as giv en b elo w . Again, f or I ⊆ [ n ] we write P I = P ∩ { x : A I x = b I } for the smallest face F ⊆ P that satisfies I ⊆ eq F . Lemma 2.11 (Isomorphism Lemma; dual formulation) . L et P = P ( A, a ) and Q = P ( B , b ) b e two p olytop es with n fac ets and dim P ≥ di m Q . If P I is a vertex = ⇒ Q I is nonempty for every set I ⊂ [ n ] then P and Q ar e of the same c ombinatorial typ e. Pr o of of Pr op osition 2.10. Sin ce Q is a simp le p olytop e, w e can find an M ≫ 0 su c h that Q ∼ = P ( B , M b − C v ) for ev ery v ∈ vert P . In p articular, if u ∈ vert Q is a v ertex w ith I = eq u then P ( B , M b − C v ) I is a v ertex. Th us, b y the du al Isomorph ism Lemma, the result follo ws . Prop osition 2.10 frees us from a d iscussion of righ t hand sides. Therefore all deformed pro d ucts hereafter are und ersto o d with a suitable right hand side. T o see that the ab ov e definition of rank r d eformed pro d ucts generalizes the (rank 1) deform ed pro du cts of Amenta & Ziegler [1 ], we recall their H -description of a deform ed p ro duct. Let P = P ( A, a ) ⊂ R d b e a p olytop e and ϕ : P → R an affin e f unctional with ϕ ( P ) ⊆ [0 , 1]. Let Q 1 , Q 2 ⊂ R e b e “normally equiv alen t” e -p olytop es, that is, combinato rially equiv alent p olytop es with the same left-hand side matrix, Q i = P ( B , b i ) f or i = 1 , 2. Th en according to [1, Thm. 3.4(iii)] the exterior r ep resen tation of ( P , ϕ ) ⊲ ⊳ ( Q 1 , Q 2 ) of the AZ-deforme d pr o duct is giv en b y ( P , ϕ ) ⊲ ⊳ ( Q 1 , Q 2 ) = n ( x, y ) ∈ R d + e : Ax ≤ a, B y ≤ b 1 − ( b 1 − b 2 ) ϕ ( x ) o 6 Prop osition 2.12. The AZ deforme d pr o duct i s a r ank 1 deforme d pr o duct. Pr o of. Let ϕ ( x ) = c T x + δ b e the affine fu nctional. Let C = ( b 1 − b 2 ) c T b e the matrix of rank at most 1 with en tr ies C ij := ( b 1 − b 2 ) i · c j . F urther, let b = b 1 − δ ( b 1 − b 2 ) and Q = P ( B , b ). No w, rewr iting the inequalit y system for ( P , ϕ ) ⊲ ⊳ ( Q 1 , Q 2 ) p ro v es the claim. 3 Neigh b orly Cubical P olytop es F or ε > 0 the int erv al I ε = { x ∈ R : ± εx ≤ 1 } is a 1-dimensional, simple p olytop e. Its p oset of nonempt y faces is the p oset on { + , − , 0 } with order relations + ≺ 0 and − ≺ 0. The signs ± represent the vertice s of the interv al with the suggestiv e n otation that ± names th e v ertices given b y ± εx = 1 w h ile 0 stands f or the uniqu e (improp er) 1-dimensional f ace. An n -fold p ro duct of in terv als giv es a combinato rial n -dimen s ional cub e C n with in equ alit y s y s tem ± 1 . . . ± ( n − k ) ± ( n − k +1) . . . ± n           ± ε . . . ± ε ± ε . . . ± ε           x ≤           1 . . . 1 1 . . . 1           . Ev ery ro w in the ab ov e system repr esen ts t wo inequalities: The i -th ro w prescrib es an u pp er and a lo w er b ound for th e v ariable x i . Left to the system are the lab els of the rows to whic h w e will refer in the follo wing. On th e lev el of p osets the facial structure is captured b y an n -fold direct pro duct of the p oset ab o v e. The nonempty faces of C n corresp ond to the elemen ts of { + , − , 0 } n with the (comp onent- wise) induced ord er relation. An ele ment γ ∈ { + , − , 0 } n represent s the unique f ace F γ with equalit y set eq F γ = { γ i i : i ∈ [ n ] } of dimension dim F γ = # { i ∈ [ n ] : γ i = 0 } . This, in particular, giv es the f -v ector as f i ( C n ) =  n i  2 n − i . The cub e, as an iterated pro d u ct of simple 1-p olyto p es, lends itself to deform ation b eneath the “diagonal” that yields, figurativ ely , a deforme d pro d u ct of interv als. In the follo wing we con- struct deformed cub es that all subscrib e to the same deformation sc h eme. T o a void cumb ersome descriptions, we fix a template for a deformed cub e. Definition 3.1 (Deformed C u b e T emplate) . F or n ≥ d ≥ 2, let G = { g 1 , . . . , g d − 1 } ⊂ R n − d b e an ordered collection of ro w v ectors and let ε > 0. W e denote by C n ( G ) a deform ed cub e with lhs m atrix A ( G ) = ( A, A ) =                ± ε 1 ± ε 1 . . . . . . ± ε 1 ± ε g 1 ± ε . . . . . . g d − 1 ± ε                . (1) Prop osition 2.10 assu r es of a su itable right h an d side such that C n ( G ) is a com binatorial n -cub e. Up to this p oint, we required ε to b e nothing b u t p ositiv e; this will b e su b ject to c hange, so on. 7 The p olytop e we are striving for is the image of C n ( G ) under pro jection. Recall that our pro jections w ill b e on to the last d co ordinates for w hic h th e vertica l bar in (1) is a remind er . W e now come to the fir st main result of this section. Theorem 3.2 (Joswig & Z iegler [6, Theorem 17]) . F or every 2 ≤ d ≤ n ther e is a cubic al d -p olytop e whose ( ⌊ d 2 ⌋ − 1) -skeleton is isomorph ic to that of an n - c ub e. Pr o of. The claim will b e established b y choosing the right deformation and verifying that all the necessary faces are strictly p reserv ed u nder p r o jectio n. Let Q b e a n eighb orly ( d − 2)-p olytop e with n − 1 v ertices in general p osition. In particular, Q has the prop ert y that ev ery subset of at most ⌊ d − 2 2 ⌋ = ⌊ d 2 ⌋ − 1 v ertices forms a f ace of Q . F or an arbitrary bu t fixed ord ering of the ve rtices, let G ∈ R ( n − 1) × ( n − d ) b e a Gale transform of Q . As the v ertices of Q are in general p osition, w e can choose a Gale transform of the form G =  I n − d G  , where G = { g 1 , . . . , g d − 1 } ⊂ R n − d is a n ordered col lection of ro w v ectors. Let C = C n ( G ) b e the d eformed cub e giv en by the template (1) with resp ect to G . W e claim that the p ro jection of C to the last d co ord inates yields the result. F or this w e pr o ve that all fac es of dimens ion up to k = ⌊ d 2 ⌋ − 1 sur viv e the pro jection. In order to do s o, w e prop ose the follo w in g strategy: W e will sho w that for an arbitrary v ertex v of C the incident faces of dimension ≤ k are retained. Consider A eq v , the firs t n − d columns of th e inequalities of (1) whic h are tigh t at v . Th e matrix is of the form A v := A eq v =                σ 1 ε 1 σ 2 ε 1 . . . . . . σ n − d ε 1 g 1 . . . g d − 1                ∈ R n × ( n − d ) (2) with σ 1 , . . . , σ n − d ∈ { + , −} . Since the v ertices of Q are in general p osition, by Prop osition 2.3, G is a configuration of v ectors in general p osition with resp ect to linear hyp erplanes. Thus, for ε > 0 sufficient ly small, A v tak e a wa y the first row is still the Gale transform of a p olyto p e com binatorially equiv alen t to Q . By Gale dualit y , this in p articular means that discardin g up to ⌊ d − 2 2 ⌋ = k rows from A v lea ve s the remaining ones p ositiv ely spann ing. No w, let F ⊂ C be a face of dimension ℓ ≤ k with v ∈ F . By the Pr o jectio n Lemma 2.5, F is strictly pr eserv ed by the p ro jection iff the ro ws of A I for I = eq F are p ositiv ely spannin g. Since C is simp le, A I is an n − ℓ ro w ed su b matrix of A eq v , that is, at most k rows h av e b een discarded from A v . Cho osing ε sufficien tly small also has th e effect that the ro ws of A v are in general position with r esp ect to lin ear h yp erp lanes. Thus, Corolla ry 2.8 vouc hes for th e fact that a ll face s of π d ( C n ( G )) arise f rom the pro jection of C n ( G ). The p olytop e π d ( C n ( G )) constructed in the cours e of the pro of dep ends on the c hoice of a neigh b orly ( d − 2)-p olytop e Q with n − 1 v ertices in general p osition, equipp ed with an ord er in g of its vertice s. In particular, th e order of the ve rtices is needed to d etermine G and th us C n ( G ). 8 Nev ertheless, by abuse of notation we will w rite C n ( Q ) f or the deformed cub e C n ( G ). W e w ill see in the next section that, in f act, th e com binatorics of π d ( C n ( Q )) is determined by the c hoice of Q and the verte x ord er . In Section 3.2, we sh o w th at the p olytop es constru cted in [6] corresp ond to the case w ere Q is a cyclic p olyto p e with the standard ve rtex ord ering. F or no w, we baptize the p olytope that we hav e constructed. Definition 3.3. F or p arameters n ≥ d ≥ 2 and a n eighb orly ( d − 2)-p olytop e Q on n − 1 ordered v ertices in general p osition, we denote the neighb orly cubic al p olytop e π d ( C n ( Q )) by NCP n,d ( Q ). Let us b riefly c omment o n the extremal c hoices of d . F or d = n , th e p olytop e NCP n,n ( Q ) is com b inatorially isomorphic to an n -cub e. The neighborly polytop e Q is then an ( n − 2)- p olytop e with n − 1 ve rtices, a simplex. F or d = 2, the p olytop e NCP n, 2 ( Q ) is a 2 n -gon and C n ( Q ) is, in fact, a r ealization of a Goldfarb cub e [3]. What m igh t strik e the reader as s trange is that the n eigh b orly p olytop e in qu estion is a 0-dimensional p olytope w ith n − 1 vertice s. The Gale tr ansform of suc h a p olytop e is giv en by the v ertices of a ( n − 2)-simplex with vertice s { e 1 , e 2 , . . . , e n − 2 , − 1 } . The pro of can b e adapted to yield a k -neighb orly cubic al p olytop e , that is, a p olytope h a ving its k -sk eleton isomorphic to that of an n -cub e. By [6 , Corollary 5], the n eigh b orliness is b ounded b y k ≤ ⌊ d 2 ⌋ − 1. In our constru ction this fact is reflected as follo ws. T he p olytop e NCP n,d ( Q ) is k -neigh b orly cu b ical iff Q is k -neigh b orly . By [15, Exercise 0.10], n eigh b orliness for ( d − 2)- p olytop es is b ounded b y ⌊ d − 2 2 ⌋ . 3.1 Com binatorial description of the neighborly cubical p olytop es W e describ e the face lattice of NCP n,d ( Q ) in terms of lexic o gr aphic triangulations of Q . W e start b y giving th e necessary b ac kgroun d on r egular sub divisions with an emph asis on lexicographic triangulations in terms of Gale transforms. Our main sources are the pap er b y Lee [8] and th e (up coming) b o ok by De Lo era et al. [2]. Let Q b e a simplicial D = d − 2 dimensional simp licial p olytope on N = n − 1 ordered v ertices. W e further assum e th at the v ertices of Q are in general p osition, i.e. all v ertex induced s ubp olyto p es are simplicial as well. Let the rows of V ∈ R N × D b e the v ertices of Q in some ordering, and let ω = ( ω 1 , . . . , ω N ) T ∈ R N b e a set of heights . Denote by V ω = ( ω , V ) ∈ R N × ( D +1) the ordered set of lifte d v ertices ( ω i , v i ) for i = 1 , . . . , N . Let a = ( ω 0 , v 0 ) ∈ R D +1 b e arbitrary with ω 0 ≫ max i | ω i | and consider the p olytop e Q ω = conv ( V ω ∪ a ). If ω 0 is s u fficien tly large, then the v ertex figure of a in Q ω is isomorphic to Q and the closed star of a in ∂ Q ω is isomorphic to that of the ap ex of a pyramid o ver Q . The anti -star (or deletion) of a in the b ound ary of Q ω , i.e. the faces of Q ω not conta ining a , constitute a p ure D -dimensional p olytopal complex Γ ω , the ω -induc e d (or ω -c oher ent ) sub division . The name “sub d ivision” stems from the fac t that the underlying set k Γ ω k is piecewise-linear homeomorphic to Q via the pr o j ection onto the last D co ordinates. The inclusion maximal p olytopes in Γ ω are called c el ls . Γ ω is called a triangulation if eve ry cel l is a D -simplex. Altering the height s ω ′ i = ω i + ℓ ( v i ) along an affine fu n ctional ℓ : Q → R lea v es the ind uced sub division unchanged. W e call a set of heigh ts normalize d if its supp ort is minimal in the corresp onding equiv alence class. Prop osition 3.4. L et ω T = ( ω 1 , . . . , ω N − D − 1 , 0 , . . . , 0) ∈ R N b e a normalize d set of heights and let G =  Id N − D − 1 G  ∈ R N × ( N − D − 1) . F or ε > 0 su fficiently smal l, the matrix G ω =  − ε ω G  ∈ R ( N +1) × ( N − D − 1) 9 with ω = ( ω 1 , . . . , ω n − d − 1 ) is a Gale tr ansform of a p olytop e c ombinatorial ly e quivalent to Q ω . Pr o of. It is easily verified that the columns of  1 1 O 1 + εω εω V  ∈ R ( N +1) × ( D +2) form a basis for the orthogonal complement of the column span of G ω . F or ε s ufficien tly small, the fi rst column is strictly p ositiv e and dehomogenizing with resp ect to this column yields the desired p olytop e. In particular, G ω enco des the com binatorial s tructure of Q as we ll as that of the ω -induced regular sub division. Consider th e tw o ind uced regular sub divisions of Q obtained b y lifting the vertex v 1 to heigh t ω 1 = ± h with h > 0 and fixing all the remaining heights to 0. I n b oth cases the lifted p olyto p e is a p yr amid o v er the p olytop e Q ′ = conv ( V \ v 1 ). F or ω 1 = − h the sub division is said to b e obtained by pul ling v 1 and its cells are pyramids o ver the remote facets of Q ′ , that is, the facets co mmon to b oth Q and Q ′ . This sub d ivision is, in fact, a triangulation since its cells are pyramids ov er ( D − 1)-simplices. The other sub division ( ω = + h ) is said to b e obtained b y pushing v 1 and its cells are p yramids o v er the n ewly created facets of Q ′ , which are again simplices, p lus one (p ossibly non-simp lex) cell that is Q ′ . The ord ering of the ve rtices of Q giv es rise to a c h ain of (su b -)p olytop es Q = Q 0 ⊃ Q 1 ⊃ · · · ⊃ Q N − D − 1 = ∆ D with Q i = conv { v i +1 , . . . , v N } simplicial D -p olyto p es. Let 1 ≤ k ≤ N − D − 1, then the k -th lexic o gr aphic triangulation Lex k Q of Q in the giv en v ertex order is the tr iangulation obtained b y pushing the fi rst k − 1 vertices in the giv en order and then pu lling th e k -th v ertex. That is to say , pushing v 1 creates a su b division of Q = Q 0 that has Q 1 as its only n on-simplex cell. Su b sequent ly , th e cell Q 1 gets replaced by a push ing sub division of Q 1 with r esp ect to v 2 , and so on. Finally , pulling v k +1 in Q k completes the triangulation. Th e follo wing lemma asserts that the ab ov e pro cedur e yields a regular sub division by giving a description in the spirit o f Prop osition 3.4. Lemma 3.5 ([8, Example 2] [12]) . L et ε > 0 and ω = ( ω 1 , ω 2 , . . . , ω N − D − 1 , 0 , . . . , 0) ∈ R N b e a set of normalize d heights satisfying | ω i +1 | ≤ ε | ω i | for al l 1 ≤ i ≤ N − D − 2 . If ε > 0 is sufficiently smal l, then G ω is a Gale tr ansform enc o ding Lex k Q for k = min { i : ω i < 0 } ∪ { n − d − 1 } . Definition 3.6. W e call the p olytop e L k ( Q ) = ˜ Q ω corresp ondin g to G ω the k -th lexic o gr aphic pyr amid of Q . According to the remarks follo w ing Prop osition 3.4, L k ( Q ) carries b oth th e combinatorics of Q as w ell as th at of Lex k Q . So every facet of L k ( Q ) is either a pyramid ov er a facet of Q or a cell of Lex k Q . W e are no w in a p osition to determine the combinato rics of NCP n,d ( Q ). T o b e more precise, we determine the lo c al com bin atorial stru cture, i.e. for an y giv en vertex w e describ e the s et of facets that con tain it. T he construction of a neighborly cubical p olytope dep ended on an order in g of the vertic es of Q , whic h w e fi x for the follo w ing theorem. Theorem 3.7. L et C = C n ( Q ) b e the deforme d cu b e with r esp e ct to Q . F urther, let v ∈ C b e an arbitr ary v ertex with eq v , given by σ ∈ { + , −} n . Then the vertex figur e of π d ( v ) in NCP n,d ( Q ) is isomorphic to L p ( Q ) for p = min { i ∈ [ n ] : σ i = + } ∪ { n − d − 1 } . 10 In p articular, the ( d − 1) -fac es of C c ontaining v that ar e pr eserve d by pr oje ction ar e in one-to-one c orr esp ondenc e to the fac ets of L p ( Q ) . Pr o of. After a su itable base transformation of (2) by means of column op erations, the firs t n − d columns of A eq v can b e assumed to b e of the form               − ω 1 − ω 2 · · · − ω n − d 1 1 . . . 1 ˜ g 1 . . . ˜ g d − 1               with ω i = ( − 1) i ε i i Y j =1 σ j By Lemma 3.5, this is a Gale transform of L k ( Q ) w ith k = p . An y generic pro jection of p olytop es π : P → P ′ = π ( P ) in duces a (co nt ra v arian t) order a nd rank p reserving map π # : F ∂ P ′ ֒ → F ∂ P . The f ace p oset of ∂ NCP n,d ( Q ) /u , the b oundary complex of the v ertex figure of u = π d ( v ) in NCP n,d ( Q ), is isomorphic to π # ( F ∂ NCP n,d ( Q ) ≥ u ), the image of the p rincipal filter of u . By the Pro jection Lemma, the image coincides with the em b ed d ing of L p ( Q ) in to th e vertex figure F ∂ C n ( Q ) ≥ v . Theorem 3.7 implies that the quotient NCP n,d ( Q ) /e with r esp ect to certain edges is isomorph ic to Q . T his obs er v ation imp lies the follo win g result. Corollary 3.8. Non-isomorphic neighb orly ( d − 2) -p olytop es Q and Q ′ yield non-isomorphic neighb orly cubic al p olytop es NCP n,d ( Q ) and NCP n,d ( Q ′ ) . Mor e over, ther e ar e at le ast as ma ny differ ent c ombinatorial typ es of d - dimensional neighb orly cubic al p olytop es as ther e ar e neighb orly simplicial ( d − 2) -p olytop es on n − 1 vertic es. The n u mb er of c om binatorial types of neighborly simplicial p olytop es is huge , according to Shemer [14]. 3.2 Neigh b orly cubical polyt op es from cyclic polyt op es In th is section we (re-)construct the neigh b orly cubical p olytop es of Joswig & Ziegler [6]. This sp ecializes the discussion in the p revious section to the case of Q a cyclic p olytop e in the standard v ertex ordering. By a thorough analysis of the lexicographic triangulations of cyclic p olytop es w e reco v er the “cub ical Gale’s ev enness criterion” of [6]. F or a treatmen t of cyclic p olytop es and their triangulations b eyond our needs we refer the r eader to [2] and [15]. The degree D moment curve is g iv en b y t 7→ γ ( t ) = ( t, t 2 , . . . , t D ) ∈ R D . F or give n pairwise distinct v alues t 1 , t 2 , . . . , t N ∈ R with N ≥ D + 1 the conv ex hull of the corresp ond ing p oints on the momen t curve Cyc D ( t 1 , . . . , t N ) = conv { γ ( t i ) : i ∈ [ N ] } is a conv ex D -dimensional p olytop e. A fund amen tal consequence of the theorem b elo w is that the com b inatorial t yp e of 11 Cyc D ( t 1 , . . . , t N ) is ind ep endent of the actual v alues t i . Therefore, we work with Cyc D ( N ) := Cyc d (1 , 2 , . . . , N ), the D -d imensional c yclic p olytop e on N v ertices in standard ord er . F or the sak e of notational con v enience later on, w e describ e its faces in terms of c haracteristic v ectors of cofaces: A v ector α ∈ { 0 , 1 } N names a coface of Cyc D ( N ) iff conv { γ ( i ) : α i = 0 } is a face of Cyc D ( N ). W e also extend the n otion of “co-” to su b divisions and, therefore, sp eak freely ab out c o c el ls . Let α ∈ { 0 , 1 } N suc h that # { j < i : α j = 0 } has the same parity for ev ery i ∈ [ N ] with α i = 1. Then α is called even or o dd according to this p arit y . Theorem 3.9 (Gale’s Ev enness Criterion [4, Sect. 4.7] [15, T hm. 0.7] [2, T h m. 6.2.6]) . A ve ctor α ∈ { 0 , 1 } N names a c ofac et of Cyc D ( N ) if and only if α has exactly D ze r o entries and is either even or o dd. As a byprod uct w e get that cyclic p olytop es are • simplicial, sin ce all facets h a ve exactly D v ertices, • in general p osition, since ev ery su bp olytop e is again cyclic, and • neigh b orly , since ev ery α ∈ { 0 , 1 } N with ≤ ⌊ D 2 ⌋ zeros can b e made to meet the ab ov e conditions b y changing ent ries 1 → 0. F r om a geometric p oint of view, the o dd and even (co) facets corresp ond to the u pp er and lo wer facets of Cyc D ( N ) with resp ect to the last co ordinate. Th is dic h otom y among the facets allo ws f or an explicit c h aracterizat ion of the (simplicial) cells of a p ushing/pulling sub division of Cyc D ( N ) with r esp ect to the first v ertex. Moreo ve r, since every v er tex indu ced subp olyto p e of Cyc D ( N ) is again cyclic an d from this we will d eriv e a complete description of the lexicographic triangulations of cyclic p olyto p es with v ertices in s tand ard ord er. T o pr epare for the pr ecise statemen t, let Q = Cyc D ( N ) = conv { v i = γ d ( i ) : i ∈ [ N ] } and Q ′ = conv { v 2 , . . . , v N } ∼ = Cyc D ( N − 1) the subp olytop e on all vertice s except the firs t. Let Γ b e the sub d ivision of Q obtained b y pulling or pushing v 1 . An y cell in Γ that con tains v 1 is a D -simplex and, therefore, let α ∈ { 0 , 1 } N b e a co cell with D + 1 zero entries and α 1 = 0. Indeed, an y such cell is a pyramid ov er a facet of Q ′ and th us α is of the form α = (0 , α ′ ) and α ′ adheres to the Gale’s eve nness criterion. The co cell α is part of a push ing or a pulling su b division of Q if and only if α is or is not a cofacet of Q . Clearly , the fir st gap in α is ev en and , hence, the parit y of the gaps of α ′ concludes the charac terization. Lemma 3.10. L et Q = Cyc D ( N ) b e a cyclic p olytop e and let L k ( Q ) b e a lexic o gr aphic pyr amid of Q . L et α ∈ { 0 , 1 } N +1 with D + 1 zer o entries and let p = min { i : α i = 0 } . Thus α is of the form α = (1 , 1 , . . . , 1 | {z } p − 1 , 0 , α ′ ) . Then α is a c ofac et of L k ( Q ) if and only i f one of the fol lowing c onditions is satisfie d: i) 1 = p and α ′ is a c ofac e t of Cyc d ( n ) . ii) 1 < p < k and α ′ is e ven. iii) p = k and α ′ is o dd. Pr o of. Ev ery facet cont aining the 0-th verte x is a pyramid o ver a facet of Q and ev ery incident facet is of the form α = (0 , α ′ ) w ith α ′ a cofacet of Q . If 2 ≤ p < k th en α n ames a co cell of th e pus h ing su b division of Q p − 1 = conv { v p , . . . , v N } with resp ect to v p and con taining v p . This, how ev er, is the case if an d on ly if α ′ is an ev en cofacet of Q p . The case p = k follo ws f rom similar considerations. 12 Setting N = n − 1 and D = d − 2 and com b ining the ab o v e description with Theorem 3.7, w e obtain the follo w ing result of Joswig & Ziegler. Theorem 3.11 (Cubical Gale’s Evenness C ondition [6]) . L et F b e a ( d − 1) -fac e of the deforme d cub e C = C n ( Q ) with Q := Cyc d − 2 ( n − 1) . L et eq F b e given by α ∈ { + , − , 0 } n and let p ≥ 1 b e the smal lest index such that α p = 0 . The fac e F pr oje cts to a fac et of NCP n,d ( Q ) if and only if α is of the form α = ( − , − , · · · , − | {z } p − 2 , σ, 0 , α ′ ) with | α ′ | = ( | α ′ p +1 | , . . . , | α ′ n | ) ∈ { 0 , 1 } n − p satisfies the or dinary Ga le’s evenness c ondition and for p > 1 one of the fol lowing c onditions holds: i) σ = − and | α ′ | is even, or ii) σ = + and | α ′ | is o dd. Pr o of. Let v ∈ F ⊂ C b e a v ertex with equalit y set β = eq v and su c h that β p = +. By Theorem 3.7, th e vertex figur e of π d ( v ) in NCP n,d ( Q ) is isomorphic to L k ( Q ), with k ∈ { p − 1 , p } . Th us F p ro jects to a facet of NCP n,d ( Q ) if and only if | α | is a cofacet of L k ( Q ). The result no w follo ws fr om Lemma 3.10 by n oting that k = p − 1 iff σ = − . 4 Deformed P ro duc ts of P olygons The pr oje cte d deforme d pr o ducts of p olygons (PDPPs) are 4-dimensional p olytop es. They were constructed in [17] b ecause o f their extremal f -v ectors: F or these p olytop es the fatness p a- rameter Φ( P ) := f 1 + f 2 − 20 f 0 + f 3 − 10 is large, getti ng arbitrarily close to 9. This paramete r, in tro duced in [16 ], is cru cial for the f -v ector theory of 4-p olyto p es. In [17] the f -v ectors of the PDPPs we re computed w ith ou t ha ving a com b inatorial characte rization of the p olytopes in r eac h. Ho wev er, th e PDPPs are ye t another instance of p ro jections of deformed pr o ducts, so the theory dev elop ed h er e giv es us a firm grip on their p rop erties. In the follo w ing w e generalize the con- struction to higher dimensions an d analyze its com binatorial stru cture using the to ols devel op ed in this pap er. In particular, a description of the facets of the PDPPs app ears for the firs t time. T o b egin with, the follo wing is a generalizatio n of Theorem 3.2. Theorem 4.1. L et m ≥ 4 b e even. F or every 2 ≤ d ≤ 2 r ther e is a d -p olytop e whose ( ⌊ d 2 ⌋ − 1) - skeleton is c ombinatorial ly isomorphic to that of an r - fold pr o duct of m -gons. Let us r emark that the pro ofs of th e results in this section can b e adapted to yield the general- izations for p r o ducts of eve n p olygons with v arying num b ers of vertice s in eac h factor. Ho we v er, the generalized results requ ir e more te c hnical and notational o verhead. Therefore, we trade generalit y in for clarit y and only give the uniform versions of the results. F or m = 4 the r -fold pro d u ct of quadrilaterals is actually a cu b e of dimension n = 2 r and thus NCP n,d ( Q ) satisfies the claims made. In the inequalit y description the quad r ilaterals can b e seen b y p airing up the in terv als indicated by th e framed submatrices b elo w: 13                       ± ε 1 ± ε 1 ± ε 1 ± ε . . . ± ε 1 ± ε 1 ± ε g 1 ± ε . . . . . . g d − 2 ± ε g d − 1 ± ε                       . W e wish to build on th is sp ecial case and therefore consider the normals of such a qu ad: P S f r a g r e p l a c e m e n t s (+ ε, 0) ( − ε, 0) (1 , + ε ) (1 , − ε ) The p olygo ns w e are heading for arise as generalizations of the ab o ve quad. F or m ≥ 4 eve n, consider the vect ors a 0 = ( − 1 , 0) a i = ( 1 , ε m − 2 i m − 2 ) for i = 1 , . . . , m − 1 as shown b elo w. F or suitable b 0 , b 1 , . . . , b m − 1 > 0, a T i x ≤ b i for i = 0 , . . . , m − 1 describ es a con v ex m -gon in the plane: P S f r a g r e p l a c e m e n t s ( − 1 , 0) = a 0 a 1 = (1 , + ε ) a 2 a m/ 2 = (1 , 0) a m − 2 a m − 1 = (1 , − ε ) 14 F or the finish in g touc h, we scale ev ery ev en-indexed inequalit y by ε , P S f r a g r e p l a c e m e n t s . . . ( − ε, 0) = εa 0 a 1 εa 2 a m / 2 − 1 εa m/ 2 a m / 2 + 1 εa m − 2 a m − 1 W e arrange the scaled normals and right hand sides into a matrix and vecto r r esp ectiv ely: A =        εa 0 a 1 εa 2 . . . a m − 1        and b =        εb 0 b 1 εb 2 . . . b m − 1        . Using these sp e cial p olygons we s et u p a template for a deformed pro d uct of p olygons (DPP). Definition 4.2 (DPP T emplate) . F or m ≥ 4 ev en and 2 r ≥ d ≥ 2, let G = { g 1 , . . . , g d − 1 } ⊂ R 2 r − d b e an ordered collecti on of row vect ors. W e d en ote by P 2 r ( G ; m ) the deformed pr o duct of p olygo ns with lhs inequ alit y s y s tem                       A 1 A . . . A 1 g 1 A . . . . . . g d − 2 g d − 1 A                       . (3) In the abov e inequalit y system, the framed blo c ks denote matrices of appropriate sizes that con tain the depicted blo ck rep eated ro w-wise m 2 times. In p articular, 1 :=        0 1 0 0 . . . 0 1 0 0        ∈ R m × 2 and 1 g 1 :=        0 · · · 0 1 g 1 . . . 0 · · · 0 1 g 1        ∈ R m × (2 r − d ) . 15 Pr o of of The or em 4.1. Let P = P 2 r ( G ; m ) b e the deformed pro d u ct of m -gons according to the DPP template (3) which is determined b y a Gale transform G =  I d − 2 G  of a neigh b orly ( d − 2)- p olytop e Q with 2 r − 1 ord ered v ertices in general p osition. Equipp ed with a suitable right hand side, the p olytope P is an iter ate d r ank 2 deforme d pr o duct of p olygons and thus combinatoria lly equiv alen t to the r -fold pro du ct of an m -gon. No w for an arbitrary vertex v of P , the matrix A eq v is of the follo wing form A eq v =                      a i 1 a ′ i 1 1 a i 2 1 a i 3 a ′ i 3 1 a i 4 . . . . . . 1 a i 2 r − d − 1 a ′ i 2 r − d − 1 1 a i 2 r − d 1 g 1 . . . g d − 1                      ∈ R 2 r × (2 r − d ) . (4) The equalit y set of a verte x v is formed by tw o cyclicly adjacen t f acets from eac h p olygon in th e pro du ct. This means, in particular, that from eac h p olygon there is an ev en and an o dd f acet present in eq v . Every suc h p air is of the form  a i ℓ a ′ i ℓ 1 a i ℓ +1  . The abs olute v alues of the diagonal ent ries are b oun ded b y ε , wh ile | a ′ i ℓ +1 | < ε 2 . Th us, pro vided that ε is sufficien tly small, the ro ws of A eq v b elo w the horizont al bar in (4 ) constitute a Gale transform of a p olytope com b inatorially equiv alen t to Q . In an alogy to the cubical ca se, we w rite P 2 r ( Q ; m ) for the deformed pro duct of m -gons with resp ect to the p olyto p e Q w ith ordered ve rtices. Definition 4.3. Th e pro of of Theorem 4.1 yields a family of p r oje cte d pr o ducts of p olygons (PDPPs) as the image PD PP 2 r,d ( Q ; m ) := π d ( P 2 r ( Q ; m )). En route to a facial description of PDPP 2 r,d ( Q ; m ), let us pause to introd uce a con venien t notation for handling pro d ucts of ev en p olygons com b inatorially that b ears certain similarities with that of 2 r -cub es, i.e. pro ducts of qu adrilaterals. F or th e ev en p olygons ab o ve , we lab el the edge with outer normal a i b y ( i, ∗ ) if i is eve n and by ( ∗ , i ) otherwise: P S f r a g r e p l a c e m e n t s ( ∗ , ∗ ) (0 , ∗ ) ( ∗ , 1) (2 , ∗ ) ( ∗ , 3) (4 , ∗ ) ( ∗ , 5) (0 , 1) (2 , 1) (2 , 3) (4 , 3) (4 , 5) (0 , 5) 16 Ev ery v ertex is incident to an ev en edge (2 i, ∗ ) and an odd edge ( ∗ , 2 i ± 1) and is lab eled by (2 i, 2 i ± 1). Fin ally , the p olygon itself gets the lab el ( ∗ , ∗ ) as the in tersection of no edges. Summing u p, the nonempty faces of an ev en m -gon are give n by P m = { (2 i, ∗ ) : 0 ≤ i < m 2 } (ev en edges) ∪ { ( ∗ , 2 i + 1) : 0 ≤ i < m 2 } (o dd edges) ∪ { (2 i, 2 i ± 1) : 0 ≤ i < m 2 } (v ertices) ∪ { ( ∗ , ∗ ) } (p olygon) with in clus ion giv en by the ord er relation ind uced by i ≺ ∗ for i ∈ { 0 , . . . , m − 1 } . Admittedly , this is neither the most natural n or the most efficie nt w a y to en co d e a p olygon com bin atorially . Ho w ev er, the follo wing remarks make up for this unusual description. S imilar to th e description of 2 r -cub es, the dimension of a face ( α 0 , α 1 ) ∈ P m is the num b er of ∗ -en tries. This carries o v er to pro du cts of m -gons, i.e. there is an order-preserving bijection b et w een the nonempt y faces of an r -fold pro d uct of m -gons and the r -fold direct pro du ct ( P m ) r with rank function dim α = # { i : α i = ∗} for α ∈ ( P m ) r . Notably most of the results (and pro ofs) from Section 3 carry o ver to this setting, with only min or mo d ifications. The key to obta ining a com binatorial description of PDPP 2 r,d ( Q ; m ) is that for a v ertex v of P 2 r ( Q ; m ) the matrix (4) again enco des a lexicographic triangulation of Q . In order to redu ce this to the case of n eigh b orly cubical p olytop es, after a suitable change of basis, the matrix A eq v is of the form                     a i 1 1 ˜ a i 2 1 a i 3 1 ˜ a i 4 . . . . . . 1 a i 2 r − d − 1 1 ˜ a i 2 r − d 1 ˜ g 1 . . . ˜ g d − 1                     α 1 α 2 α 3 α 4 . . . α 2 r − d − 1 α 2 r − d α 2 r − d + 1 α 2 r − d + 2 . . . α 2 r (5) The en tries ab o v e the diagonal of ones remain to b e of order ε . T o determine the signs of the en tries, w hic h will d etermine the lexicographic triangulation, let u s in v estigate the lo c al change of the matrix under th e change of basis. In the ab o ve com binatorial mo d el for ev en m -gons, the vertex v is identified with a v ector α = ( α 1 , α 2 ; α 3 , . . . ; α 2 r − 1 , α 2 r ) ∈ ( P m ) r , which corr esp onds to eq v as ind icated. The f ollo wing table, wh ic h is easily established given the coord inates of the n ormals, sum m arizes the p ossible sign patterns in terms of α . ( α i , α i +1 ) (0 , 1) (0 , m − 1) (2 k, 2 k − 1) (2 k, 2 k + 1)  a i 1 ˜ a i +1   − ε 1 + ε   − ε 1 − ε   + ε 1 + 2 ε m − 2   + ε 1 − 2 ε m − 2  ( σ i , σ i +1 ) ( − , +) ( − , − ) (+ , +) (+ , − ) W e use the last r o w, w hic h gathers sign patterns from the diagonal, to define the map Φ : { ( α 1 , α 2 ) ∈ P m : α vertex } → { + , − , 0 } 2 17 with Φ( α 1 , α 2 ) := ( σ 1 , σ 2 ) according to the table. Sin ce the face lattice of a con vex p olytop e is atomic, it is easy to see f r om the definition that Φ : P m → { + , − , 0 } 2 extends to an ord er - and rank-preserving map fr om the face p oset of an ev en m -gon to that of an 2-cub e. T he map can b e thought of as a folding map: P S f r a g r e p l a c e m e n t s Φ ++ + − −− − + (0 , 5) (0 , 1) (2 , 1) (2 , 3) (4 , 3) (4 , 5) The induced map Φ : ( P m ) r → { + , − , 0 } 2 r maps faces of P 2 r ( k ; m ) that are strictly pr eserved under π d to sur viving faces of C 2 r ( Q ). Ph rased differentl y the follo wing diagram comm utes on the lev el of faces: P n,r ( Q ) Φ − − − − − − − − − − → C 2 r ( Q )   y π d   y π d PDPP 2 r,d ( Q ; m ) Φ − − − − − − − − − − → NCP d ( Q ) . Prop osition 4.4. L et n = 2 r and let P = P n ( Q ; m ) and C = C n ( Q ) b e the deforme d cub e and the pr o duct of m -gons of dimension n = 2 r with r esp e ct to a neighb orly ( d − 2) -p olytop e Q on n − 1 or der e d vertic es. L et v ∈ P b e a vertex with eq v r epr esente d by α ∈ ( P m ) r and let u ∈ C b e the vertex c orr esp onding to Φ( α ) ∈ { + , −} n . Then Φ induc e s an isomorphism of the vertex figur es PDPP n,d ( Q ; m ) /π d ( v ) and NCP n ( Q ) /π d ( u ) . Pr o of. As consisten t with the main theme in this artic le, consider the first n − d = 2 r − d co ordinates of the inequalities from b oth P and C th at are tigh t at v and u , resp ectiv ely . A v ( P ) A u ( C )                 a i 1 1 ˜ a i 2 . . . . . . 1 a i n − d − 1 1 ˜ a i n − d 1 ˜ g 1 . . . ˜ g d − 1                                 σ i 1 ε 1 σ i 2 ε . . . . . . 1 σ i n − d − 1 ε 1 σ i n − d ε 1 g 1 . . . g d − 1                 In b oth matrices, the en tries on the secondary diagonal are arbitrary small and the map Φ assures th at corresp onding entries h a ve equal sign. By Lemma 3.5, b oth A v ( P ) and A u ( C ) are Gale transforms that enco de the same lexicographic pyramid L k ( Q ). The result now follo ws by observing that a face β  α of P is strictly preserved if and only if | β | is a coface of L k ( Q ) and | Φ( β ) | = | β | . 18 This prop osition mak es w a y for the com b inatorics of the pro jected deformed pro d ucts asso ciated with arb itrary simp licial neigh b orly p olytop es. Theorem 4.5 (Combinato rial Description of the PDPPs) . L et P = P 2 r ( Q ; m ) b e a deforme d pr o duct of m -gons with r esp e ct to Q and let v ∈ P b e an arbitr ary vertex with eq v = α ∈ ( P m ) r . Then the vertex figur e of π d ( v ) in PDPP 2 r,d ( Q ; m ) is isomorph ic to L p ( Q ) for p = min { i ∈ [2 r ] : Φ( α ) i = −} ∪ { 2 r − d − 1 } . In p articular, the ( d − 1) -fac es of P c ontaining v that ar e pr e se rve d by pr oje ction ar e in one-to-one c orr esp ondenc e to the fac ets of L p ( Q ) . As for the neighborly cubical p olytop es, via S hemer’s work [14] this result imp lies a great richness of com binatorial t yp es for the p ro jected pro du cts of p olygo ns. In the sp ecial case when Q is a cyclic p olytop e w ith vertice s in standard order, w e get a very explicit Gale’s evenness -t yp e criterion f or the pr o jected p ro ducts of p olygons. Corollary 4.6 (Com binatorial Description of the standard PDPPs) . L et F ⊂ P = P 2 r ( Q ; m ) b e a ( d − 1) -fac e with Q = Cyc d − 2 (2 r − 1) and let β ∈ ( P m ) r c orr esp ond to eq F . Then F pr oje cts to a fac et of PDPP 2 r,d ( Q ; m ) if and only if Φ ( β ) satisfies the cub ic al Gale’s evenness criterion. References [1] N . Ament a and G. M. Ziegler , Deforme d pr o ducts and max imal shadows of p olytop es , in Ad- v ances in Discre te a nd Computatio nal Geometry , B. Chaz elle, J. Goodman, and R. Pol lack , eds., vol. 22 3 of Contempor a ry Mathematics, Amer. Ma th. So c., P rovidence RI, 199 8, pp. 57 –90. 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