#### Construction and Analysis of Projected Deformed Products

Discrete Comput Geom
Construction and Analysis of Projected Deformed Products
Raman Sanyal 0 1
Günter M. Ziegler 0 1
Dedicated to Victor Klee. 0 1
0 G.M. Ziegler ( ) Inst. Mathematics , MA 6-2, TU Berlin, 10623 Berlin , Germany
1 R. Sanyal Department of Mathematics, UC Berkeley , Berkeley, CA 94720-3840 , USA
We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and the analysis of deformed products such that specified faces (e.g., all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d − 2)-polytope Q on n − 1 vertices, we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d -polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs) which have a projection to d -space that retains the complete ( d2 − 1)-skeleton. In both cases the combinatorial structure of the images under projection is determined by the neighborly simplicial polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig and Ziegler (Discrete Comput. Geom. 24:325-344, 2000) as well as of the projected deformed products of polygons announced by Ziegler (Electron. Res. Announc. Am. Math. Soc. 10:122-134, 2004), a family of 4-polytopes whose “fatness” gets arbitrarily close to 9.
1 Introduction
Some remarkable geometric effects can be achieved for projections of
“suitablydeformed” high-dimensional simple polytopes. This includes the Klee–Minty
cubes [
8
], the Goldfarb cubes [
4
], and many other exponential examples for
variants of the simplex algorithm, but also the “neighborly cubical polytopes” first
constructed by Joswig and Ziegler [
7
]. A geometric framework for “deformed product”
constructions was provided by Amenta and Ziegler [
1
].
Here we introduce a generalized deformed products construction. In terms of this
construction, the previous version by Amenta and Ziegler concerned deformed
products of rank 1. The new construction is presented in matrix version (that is, as an
H-polytope). Iterated deformed products are thus given by lower-triangular block
matrices, where the blocks below the diagonal do not influence the combinatorics of
the product (for suitable right-hand sides).
The deformed products P are constructed in order to provide interesting images
after an affine projection π : P → π(P ). The deformations we are after are designed
so that certain classes of faces of the deformed product P , e.g., all the k-faces, are
“preserved” by a projection to some low-dimensional space, i.e., mapped to faces
of π(P ). In the combinatorially convenient situation, the faces in question are strictly
preserved by the projection; we give a linear algebra condition that characterizes the
faces that are strictly preserved (Projection Lemma 2.5). We also identify a situation
where 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 by the Projection Lemma may be translated via a
nonstandard application of Gale duality [5, Sect. 6.3], [16, Sect. 6] into conditions about the
combinatorics of an auxiliary polytope Q.
As an instance of this set-up, we show how neighborly cubical d-polytopes arise
from projections of a deformed n-cube where all the ( d2 − 1)-faces are preserved
by the projection. The precise form of the matrix representation of the n-cube and
the combinatorics of the resulting polytopes are dictated via Gale duality by a
neighborly simplicial (!) (d − 2)-polytope with n − 1 vertices. As special cases, we obtain
the neighborly cubical polytopes first obtained by Joswig and Ziegler [
7
] and also
geometric realizations for neighborly cubical spheres as described by Joswig and
Rörig [
6
] and originally by Babson, Billera, and Chan [
2
].
Finally, we construct and analyze projected deformed products of (even) polygons
(PDPP polytopes) as the images of a deformed product of r even polygons projected
to Rd . The projection is designed to strictly preserve all the ( d2 − 1)-faces (and
additional d2 -faces if d is even). This produces in particular the two-parameter family of
four-dimensional polytopes from [
18
], for which the “fatness” parameter introduced
in [
17
] gets as large as 9 − ε. We present a new construction (drastically simplified
and systematized) and a complete combinatorial description of these polytopes.
This work is based on the Diploma thesis [
13
]; see also the research
announcements in [
18
] and [
19
]. The “wedge product” polytopes of Rörig and Ziegler [
12
]
provide another interesting instance of “deformed high-dimensional simple polytopes.”
A further analysis shows that the neighborly cubical polytopes, the PDPP polytopes,
and the wedge products do exhibit a wealth of interesting polyhedral surfaces,
including the “surfaces of unusually high genus” by McMullen, Schulz, and Wills [
10
],
and equivelar surfaces of type (p, 2q). Topological obstructions that prevent a
suitable projection of “deformed products of odd polygons,” or of the wedge product
polytopes, will be presented by Rörig and Sanyal [
11
].
2 Basics
In this section we recall basic properties and notation about the main objects of this
paper, convex polytopes. Readers new to the country of polytopia will find useful
information in the well-known travel guides [
5, 16
], while the frequent visitors might
wish to skim the section for possibly nonstandard notation.
One of the main messages this article tries to convey is that it pays off to work
with polytopes in explicit coordinates (matrix representation). Classically, there
are two fundamental ways of viewing a polytope in coordinates: the interior or
V -representation, and the exterior or H-representation. For V -polytopes “with few
vertices,” Perles [5, Chap. 6] had developed Gale duality as a powerful tool. In this
article, we will apply Gale duality for the analysis of projected simple H-polytopes.
The basics for this will be developed in this section.
2.1 Polytopes in Coordinates
For the rest of the section, let P ⊂ Rd be a full-dimensional polytope. In its interior
or V -presentation, P = conv V is given as the convex hull of a finite point set V =
{v1, . . . , vm} ⊂ Rd , and V is inclusion-minimal with respect to this property. The
elements of V are called vertices, with notation vert P = V . For a nonempty subset
I ⊆ [m] = {1, . . . , m}, the set VI = {vi : i ∈ I } forms a face of P if there is a linear
functional : Rd → R such that attains its maximum over P on F = conv V . The
dimension dim F is the dimension of its affine span. The empty set is also a face of P
of dimension −1. The collection F P of all faces of P , ordered by inclusion, is a
graded, atomic, and coatomic lattice with dim + 1 as its rank function. We denote by
F ∂ P := F P \ {P } the face poset of the boundary of P . We say that two polytopes are
of the same combinatorial type if their face lattices are isomorphic as abstract posets.
A polytope P is simplicial if small perturbations applied to the vertices do not alter
its combinatorial type. Equivalently, every k-face of P (k < dim P ) is the convex hull
of exactly k + 1 vertices. The quotient P /F of P by a face F is a polytope with face
lattice isomorphic to F P≥F = {G ∈ F P : F ⊆ G}. If F = {v} is a vertex, then P /v
is called a vertex figure at v.
The polytope P is given in its exterior or H-presentation if P is the intersection
of finitely many halfspaces. That is, if there are (outer) normals a1, . . . , an ∈ Rd and
displacements b1, . . . , bn ∈ R such that
P =
n
i=1
x ∈ Rd : aiTx ≤ bi ,
where we assume that the collection of normals is irredundant, thus
discarding any one of the halfspaces changes the polytope. The hyperplanes Hi =
{x ∈ Rd : aiTx = bi } are said to be facet defining; the corresponding (d − 1)-faces
Fi = P ∩ Hi are called facets. More compactly, we think of the normals ai as the
rows of a matrix A ∈ Rn×d and, with b ∈ Rn accordingly, write
P = P (A, b) = x ∈ Rd : A x ≤ b .
For any subset F ⊆ P , let eq F = {i ∈ [n] : F ⊂ Hi } ⊆ [n] be its equality set. Clearly,
F ⊆ i∈eq F Fi ; in case of equality, the set F is a face of P . Denote by AI the sub
matrix of A induced by the row indices in I ⊆ [n]. Thus any face F is given by
F = P ∩ {x : AI x = bI } for I = eq F . The collection of equality sets of faces ordered
by reverse inclusion is isomorphic to F P . The polytope P is simple if its
combinatorial type is stable under small perturbations applied to the bounding hyperplanes.
Equivalently, every nonempty face F is contained in no more than |eq F | = d − dim F
facets.
2.2 Gale Duality
Let P ⊂ Rd be a d -polytope, and let the rows of V ∈ Rm×d be the m vertices of P .
Denote by V hog = (V , 1) ∈ Rm×(d+1) the homogenization of V . The column span of
V is a (d + 1)-dimensional linear subspace. Choose G ∈ Rm×(m−d−1) such that the
columns form a basis for the orthogonal complement. Any such basis, regarded as
an ordered collection of m row vectors, is called a Gale transform of P . It is unique
up to linear isomorphism and, by the reverse process, characterizes V hog, again up
to linear isomorphism. So it determines P only up to a projective transformation.
However, the striking feature of Gale transforms is that its combinatorial properties
are, in a precise sense, dual to those of P ; this correspondence goes by the name of
Gale duality.
In order to state and work with Gale duality, we introduce some concepts and
notation. As before, we write VI for the subset of the rows of V indexed by I ⊆ [m].
A subset I ⊂ [m] names a coface of P if the complement V[m]\I is the vertex set of a
face of P .
Definition 2.1 A collection of vectors G = {g1, g2, . . . , gm} ⊂ Rk is positively
dependent if there are numbers λ1, λ2, . . . , λm > 0 such that λ1g1 + · · · + λmgm = 0. It
is positively spanning if in addition G is of full rank k.
“Being positively spanning” is, like “being spanning,” an open condition, i.e.,
preserved under (sufficiently small) perturbations of the elements of G. This, however,
is not true for “being positively dependent;” consider, e.g., {g, −g} for g ∈ Rk , g = 0,
k > 1.
Theorem 2.2 Gale duality Let P = conv V be a polytope and G a Gale transform
of P . Let I ⊂ [m]. Then I names a coface of P if and only if GI is positively
dependent.
In light of Gale duality, the preceding theorem implies that, for a general polytope,
not every subset of the vertex set of a face necessarily forms a face. This, however, is
true for simplicial polytopes and, in fact, characterizes them. A still stronger condition
is satisfied if no d + 1 vertices of a d-polytope lie on a hyperplane, that is, if the
vertices are in general position with respect to affine hyperplanes. This translates
into Gale diagrams as follows.
Proposition 2.3 Let P ⊂ Rd be a polytope and G ⊂ Rk a Gale transform of P . Then
P is simplicial with vertices in general position if and only if the rows of G are in
general position with respect to linear hyperplanes, that is, if any k vectors of G are
linearly independent.
2.3 Faces Strictly Preserved by a Projection
Projections are fundamental in polytope theory: Every polytope on n vertices is the
image of an (n − 1)-simplex under an affine projection. This in particular says that
the analysis of the images of polytopes under projection is as difficult as the
general classification of all combinatorial types of polytopes. The problem is that a
k-face F ⊂ P can behave in various ways under projection: It can map to a k-face,
or to part of a k-face, or to a lower-dimensional face of π(P ). Even if it maps to a
k-face π(F ) ⊂ π(P ), there may be other k-faces of P that map 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, we restrict our attention in the following to the most
convenient situation of faces that are “strictly preserved” by a projection.
Definition 2.4 (Strictly preserved faces [
18
]) Let P be a polytope and Q = π(P )
the image of P under an affine projection π : P → Rd . A nonempty face F of P is
(strictly) preserved by π if:
(i) π(F ) is a face of Q combinatorially equivalent to F , and (preserved face)
(ii) the preimage π −1(π(F )) is F (strictly preserved).
Since in the following we will be concerned exclusively with the analysis of
strictly preserved faces, we will generally drop the modifier “strictly” starting now.
The following lemma gives an algebraic way to read off the preserved faces from
a polytope in exterior presentation. Every affine projection π : Rn → Rd factors as
an affine transformation followed a projection πd : Rn−d × Rd → Rd that deletes the
first n − d coordinates, that is, πd (x, x) = x for all (x, x) ∈ Rn−d × Rd . Therefore,
we will focus on the projections πd “to the last d coordinates.” For a polytope P =
P (A, b) ⊂ Rn in exterior presentation, the projection map πd naturally partitions the
columns of A as A = (A|A).
Lemma 2.5 (Projection Lemma: matrix version) Let P = P (A, b) ⊂ Rn be a
polytope, F a nonempty face of P , and I = eq F the index set of the inequalities that
are tight at F . Then F is preserved by the projection πd : P → Rd to the last d
coordinates if and only if the rows of AI are positively spanning.
The proof makes use of the following geometric version of the Farkas Lemma.
Lemma 2.6 [16, Sect. 1.4] Let P = P (A, b) be a polytope and F ⊆ P a nonempty
face. For a linear functional (x) = cx, we denote by P the nonempty face of P on
which attains its maximum. The linear function singles out F , that is, P = F , if
and only if c is a strictly positive linear combination of the rows of Aeq F .
Proof of Lemma 2.5 We split the proof into two parts.
Claim 1 F˜ = πd (F ) is a face of P˜ = πd (P ) with πd−1(F˜ ) ∩ P = F iff AI is
positively dependent.
By Lemma 2.6 the rows of AI are positively dependent if and only if there is some
c ∈ Rd such that the linear function (x) := (0, c) x = c x satisfies P = F . Rewriting
= h ◦ πd with h(x) = c x, we see that such 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-)polytope in its own right, then F˜ = πd (F ) is
combinatorially equivalent to F if and only if AI has full row rank.
The polytopes F and F˜ are combinatorially equivalent iff they are affinely
isomorphic. This happens if and only if the linear map πd is injective restricted the linear
space L = {x : AI x = 0}, which is parallel to aff F = {x : AI x = bI }, the affine hull
of F . Now, πd |L is injective iff ker πd ∩ L ∼= {x : AI x = 0} is trivial.
See [
14
] for a proof in a different wording.
Lemma 2.5 allows us to guarantee that in certain situations every single k-face is
preserved by a projection π : P → π(P ). Then, however, we want to also see that
π(P ) has no other k-face than those induced by the projection. This will be argued
via the following lemma.
Lemma 2.7 Let P = P (A, b) ⊂ Rn be an n-polytope such that, for every vertex
v ∈ vert P , the rows of the matrix Aeq v are in general position with respect to linear
hyperplanes. Then every proper face of P is either preserved under πd , or its image
under πd is not a face of πd (P ).
Proof If G ⊂ Rk is a set of at least k vectors in general position with respect to linear
hyperplanes, then dim span G ≥ min{|G |, k} for every subset G ⊆ G. In particular,
every positively dependent subset is positively spanning.
Let F ⊂ P be a proper face. From the proof of Lemma 2.5 it follows that πd (F ) is
a face iff Aeq F is positively dependent. Let v ∈ vert P be a vertex with v ∈ F . Then
Aeq F ⊆ Aeq v , and Aeq v ⊂ Rn−d is a set of at least n vectors in general position with
respect to linear hyperplanes.
Corollary 2.8 If all k-faces of P are preserved by the projection π : P → π(P ),
then all k-faces of π(P ) arise as images of k-faces of P .
Proof For any k-face G ⊆ π(P ), we know that G = πd −1(G) is a face of P of
dimension dim G ≥ k. Now if F ⊆ G is any k-face of G, then by Lemma 2.7 either
F is preserved, and we get πd (F ) = G, or F is not mapped to a face. The latter case
cannot arise here.
2.4 Generalized Deformed Products
The orthogonal product P × Q ⊂ Rd+e of a d-polytope P = P (A, a) ⊂ Rd and an
e-polytope Q = P (B, b) ⊂ Re is given in inequality description by the block
diagonal system
Ax
≤ a,
By ≤ b.
We get a deformed product (with the combinatorial structure of the orthogonal
product) if we generalize this into a block lower-triangular system, provided that Q is
simple and that we rescale the right-hand side of the system suitably.
Definition 2.9 (Rank r deformed product) Let P = P (A, a) ⊂ Rd be a d-polytope
and Q = P (B, b) ⊂ Re a simple e-polytope with A ∈ Rk×d and B ∈ Rn×e. Let
C ∈ Rn×d be an arbitrary matrix of rank r , and let M 0 be large. The rank r deformed product P C Q ⊂ Rd+e of P and Q with respect to C is given by
Ax ≤ a,
Cx + By ≤ Mb,
that is
A
C B
x
y =
Mab .
Proposition 2.10 Let P = P (A, a) ⊂ Rd be a d-polytope, Q = P (B, b) ⊂ Re a
simple e-polytope, P C Q their deformed product, and M > 0 the parameter involved.
If M is sufficiently large (depending on B, b, and C), then P C Q and P × Q are
combinatorially equivalent.
Our proposition may also be obtained from the Isomorphism Lemma [1,
Lemma 2.4] that was applied by Amenta and Ziegler to prove the corresponding statement
for (rank 1) deformed products. However, we use it in a dual form as given below.
Again, for I ⊆ [n], we write PI = P ∩ {x : AI x = bI } for the smallest face F ⊆ P
that satisfies I ⊆ eq F .
Lemma 2.11 (Isomorphism Lemma; dual formulation) Let P = P (A, a) and
Q = P (B, b) be two polytopes with n facets and dim P ≥ dim Q. If
PI is a vertex
=⇒
QI is nonempty
for every set I ⊂ [n], then P and Q are of the same combinatorial type.
Proof of Proposition 2.10 Since Q is a simple polytope, we can find M 0 such
that Q ∼= P (B, Mb − Cv) for every v ∈ vert P . In particular, if u ∈ vert Q is a vertex
with I = eq u, then P (B, Mb − Cv)I is a vertex. Thus, by the dual Isomorphism
Lemma, the result follows.
Proposition 2.10 frees us from a discussion of right-hand sides. Therefore all
deformed products hereafter are understood with a suitable right-hand side.
To see that the above definition of rank r deformed products generalizes the
(rank 1) deformed products of Amenta and Ziegler [
1
], we recall their H-description
of a deformed product. Let P = P (A, a) ⊂ Rd be a polytope and ϕ : P → R an
affine functional with ϕ(P ) ⊆ [
0, 1
]. Let Q1, Q2 ⊂ Re be “normally equivalent”
epolytopes, that is, combinatorially equivalent polytopes with the same left-hand side
matrix, Qi = P (B, bi ) for i = 1, 2. Then, according to [1, Theorem 3.4(iii)], the
exterior representation of (P , ϕ) (Q1, Q2) of the AZ-deformed product is given by
(P , ϕ)
(Q1, Q2) = (x, y) ∈ Rd+e : Ax ≤ a, By ≤ b1 − (b1 − b2)ϕ(x) .
Proposition 2.12 The AZ-deformed product is a rank 1 deformed product.
Proof Let ϕ(x) = cTx + δ be the affine functional. Let C = (b1 − b2)cT be the
matrix of rank at most 1 with entries Cij := (b1 − b2)i · cj . Further, let b = b1 −
δ(b1 − b2) and Q = P (B, b). Now, rewriting the inequality system for (P , ϕ)
(Q1, Q2) proves the claim.
3 Neighborly Cubical Polytopes
For ε > 0, the interval Iε = {x ∈ R : ±εx ≤ 1} is a one-dimensional, simple polytope.
Its poset of nonempty faces is the poset on {+, −, 0} with order relations + ≺ 0 and
− ≺ 0. The signs ± represent the vertices of the interval with the suggestive notation
that ± names the vertices given by ±εx = 1, while 0 stands for the unique
(improper) one-dimensional face. An n-fold product of intervals gives a combinatorial
n-dimensional cube Cn with inequality system
±1 ⎛ ±ε
.
.
. ⎜⎜
±(n−k) ⎜⎜
±(n−k+1). ⎜⎜⎜
.. ⎜
⎝
±n
±ε
±ε
. . .
±ε
⎞ ⎛ 1 ⎞
.
⎟⎟⎟ ⎜⎜⎜ 1.. ⎟⎟⎟
⎟⎟⎟⎟⎟ x ≤ ⎜⎜⎝⎜⎜⎜ 1... ⎟⎟⎟⎟⎟ .
⎠ ⎠
1
Every row in the above system represents two inequalities: The ith row prescribes an
upper and a lower bound for the variable xi . Left to the system are the labels of the
rows to which we will refer in the following.
On the level of posets the facial structure is captured by an n-fold direct product of
the poset above. The nonempty faces of Cn correspond to the elements of {+, −, 0}n
with the (component-wise) induced order relation. An element γ ∈ {+, −, 0}n
represents the unique face Fγ with equality set eq Fγ = {γi i : i ∈ [n]} of dimension
dim Fγ = #{i ∈ [n] : γi = 0}. This, in particular, gives the f -vector as fi (Cn) =
ni 2n−i .
The cube, as an iterated product of simple 1-polytopes, lends itself to deformation
beneath the “diagonal” that yields, figuratively, a deformed product of intervals. In
the following we construct deformed cubes that all subscribe to the same deformation
scheme. To avoid cumbersome descriptions, we fix a template for a deformed cube.
Definition 3.1 (Deformed Cube Template) For n ≥ d ≥ 2, let G = {g1, . . . , gd−1} ⊂
Rn−d be an ordered collection of row vectors, and let ε > 0. We denote by Cn(G) the
deformed cube with lhs matrix
⎛ ±ε
1
±ε
1
. . .
±ε
⎞
⎟⎟⎟⎟⎟
⎟⎟ .
⎟⎟⎟⎟⎟⎟
⎠
(1)
Proposition 2.10 assures of a suitable right-hand side such that Cn(G) is a
combinatorial n-cube. Up to this point, we required ε to be nothing but positive; this will
be subject to change, soon.
The polytope we are striving for is the image of Cn(G) under projection. Recall
that our projections will be onto the last d coordinates for which the vertical bar in
(1) is a reminder.
We now come to the first main result of this section.
Theorem 3.2 (Joswig and Ziegler [7, Theorem 17]) For every 2 ≤ d ≤ n, there is a
cubical d-polytope whose ( d2 − 1)-skeleton is isomorphic to that of an n-cube.
Proof Let Q be a neighborly simplicial (d − 2)-polytope with n − 1 vertices in
general position. In particular, Q has the property that every subset of at most
d −22 = d2 − 1 vertices forms a face of Q. For an arbitrary but fixed ordering of
the vertices, let G ∈ R(n−1)×(n−d) be a Gale transform of Q. As the vertices of Q are
in general position, we can choose a Gale transform of the form G = In−d , where
G
G = {g1, . . . , gd−1} ⊂ Rn−d is an ordered collection of row vectors. Let C = Cn(G)
be the deformed cube given by the template (1) with respect to G.
We claim that the projection of C to the last d coordinates yields the result. For
this, we prove that all faces of dimension up to k = d2 − 1 survive the projection. In
order to do so, we propose the following strategy: We will show that, for an arbitrary
vertex v of C, the incident faces of dimension ≤ k are retained.
Consider Aeq v , the first n − d columns of the inequalities of (1) which are tight
at v. The matrix is of the form
(2)
g1
.
.
.
gd−1
⎞
. . .
. .
⎟
⎟
⎟
⎟
⎟
. σn−d ε ⎟⎟⎟ ∈ Rn×(n−d)
1 ⎟
⎟
⎟
⎟
⎟
⎟
⎠
with σ1, . . . , σn−d ∈ {+, −}.
Since the vertices of Q are in general position, by Proposition 2.3, G is a
configuration of vectors in general position with respect to linear hyperplanes. Thus, for
ε > 0 sufficiently small, Av take away the first row is still the Gale transform of a
polytope combinatorially equivalent to Q. By Gale duality, this in particular means
that discarding up to d −22 = k rows from Av leaves the remaining ones positively
spanning.
Now, let F ⊂ C be a face of dimension ≤ k with v ∈ F . By the Projection
Lemma 2.5, F is strictly preserved by the projection iff the rows of AI for I = eq F
are positively spanning. Since C is simple, AI is an (n − )-rowed submatrix of Aeq v ,
that is, at most k rows have been discarded from Av .
Choosing ε sufficiently small also has the effect that the rows of Av are in general
position with respect to linear hyperplanes. Thus, Corollary 2.8 vouches for the fact
that all faces of πd (Cn(G)) arise from the projection of Cn(G).
The polytope πd (Cn(G)) constructed in the course of the proof depends on the
choice of a neighborly simplicial (d − 2)-polytope Q with n − 1 vertices in general
position, equipped with an ordering of its vertices. In particular, the order of the
vertices is needed to determine G and thus Cn(G). Nevertheless, by abuse of notation
we will write Cn(Q) for the deformed cube Cn(G). We will see in the next section
that, in fact, the combinatorics of πd (Cn(Q)) is determined by the choice of Q and
the vertex order. In Sect. 3.2, we show that the polytopes constructed in [
7
]
correspond to the case where Q is a cyclic polytope with the standard vertex ordering.
According to Joswig and Rörig [
6
], they have the same combinatorial types as the
spheres originally constructed by Babson, Billera, and Chan [
2
]. For now, we baptize
the polytope that we have constructed.
Definition 3.3 For n ≥ d ≥ 2 and a neighborly simplicial (d − 2)-polytope Q on
n − 1 ordered vertices in general position, we denote the neighborly cubical polytope
πd (Cn(Q)) by NCPn,d (Q).
Let us briefly comment on the extremal choices of d. For d = n, the polytope
NCPn,n(Q) is combinatorially isomorphic to an n-cube. The neighborly
simplicial polytope Q is then an (n − 2)-polytope with n − 1 vertices, a simplex. For
d = 2, the polytope NCPn,2(Q) is a 2n-gon, and Cn(Q) is, in fact, a realization of a
Goldfarb cube [
4
]. Note that for this the “neighborly simplicial polytope” in question
is a 0-dimensional polytope consisting of n − 1 points. The Gale transform of such
a polytope/configuration is given by the vertices of an (n − 2)-simplex with vertices
{e1, e2, . . . , en−2, −1}.
The proof can be adapted to yield a k-neighborly cubical polytope, that is, a
polytope having its k-skeleton isomorphic to that of an n-cube. By [7, Corollary 5], the
neighborliness is bounded by k ≤ d2 − 1. In our construction this fact is reflected
as follows. The polytope NCPn,d (Q) is k-neighborly cubical iff Q is k-neighborly
simplicial. By [16, Exercise 0.10], neighborliness for (d − 2)-polytopes is bounded
by d−2 .
2
3.1 Combinatorial Description of the Neighborly Cubical Polytopes
We describe the face lattice of NCPn,d (Q) in terms of lexicographic triangulations
of Q. We start by giving the necessary background on regular subdivisions with
an emphasis on lexicographic triangulations in terms of Gale transforms. Our main
sources are the paper by Lee [
9
] and the (upcoming) book by De Loera et al. [
3
].
Let Q be a simplicial D-dimensional simplicial polytope for D = d − 2 on
N = n − 1 ordered vertices. We further assume that the vertices of Q are in general
position, i.e., all vertex-induced subpolytopes are simplicial as well. Let the rows of
V ∈ RN×D be the vertices of Q in some ordering, and let ω = (ω1, . . . , ωN )T ∈ RN
be a set of heights. Denote by V ω = (ω, V ) ∈ RN×(D+1) the ordered set of lifted
vertices (ωi , vi ) for i = 1, . . . , N . Let a = (ω0, v0) ∈ RD+1 be arbitrary with ω0
maxi |ωi | and consider the polytope Qω = conv (V ω ∪ a). If ω0 is sufficiently large,
then the vertex figure of a in Qω is isomorphic to Q, and the closed star of a in
∂Qω is isomorphic to that of the apex of a pyramid over Q. The anti-star (or
deletion) of a in the boundary of Qω, i.e., the faces of Qω not containing a, constitute
a pure D-dimensional polytopal complex Γω, the ω-induced (or ω-coherent)
subdivision. The name “subdivision” stems from the fact that the underlying set Γω is
piecewise-linear homeomorphic to Q via the projection onto the last D coordinates.
The inclusion maximal polytopes in Γω are called cells. Γω is called a triangulation if
every cell is a D-simplex. Altering the heights ωi = ωi + (vi ) along an affine
functional : Q → R leaves the induced subdivision unchanged. We call a set of heights
normalized if its support is minimal in the corresponding equivalence class.
Proposition 3.4 Let ωT = (ω1, . . . , ωN−D−1, 0, . . . , 0) ∈ RN be a normalized set of
threiixghts, and let G = IdNG−D−1 ∈ RN×(N−D−1). For ε > 0 sufficiently small, the
maGω =
−εω
G
∈
R(N+1)×(N−D−1)
with ω = (ω1, . . . , ωn−d−1) is a Gale transform of a polytope combinatorially
equivalent to Qω.
Proof It is easily verified that the columns of
O
V
∈
R(N+1)×(D+2)
form a basis for the orthogonal complement of the column span of Gω. For ε
sufficiently small, the first column is strictly positive, and dehomogenizing with respect
to this column yields the desired polytope.
In particular, Gω encodes the combinatorial structure of Q and that of the
ω-induced regular subdivision.
Consider the two induced regular subdivisions of Q obtained by lifting the
vertex v1 to height ω1 = ±h with h > 0 and fixing all the remaining heights to 0. In
both cases the lifted polytope is a pyramid over the polytope Q = conv (V \ v1).
For ω1 = −h, the subdivision is said to be obtained by pulling v1, and its cells are
pyramids over the remote facets of Q , that is, the facets common to both Q and Q .
This subdivision is, in fact, a triangulation since its cells are pyramids over (D − 1)
simplices. The other subdivision (ω = +h) is said to be obtained by pushing v1, and
its cells are pyramids over the newly created facets of Q , which are again simplices,
plus one (possibly nonsimplex) cell that is Q .
The ordering of the vertices of Q gives rise to a chain of (sub-)polytopes
Q = Q0 ⊃ Q1 ⊃ · · · ⊃ QN−D−1 = D with Qi = conv {vi+1, . . . , vN } simplicial
D-polytopes. Let 1 ≤ k ≤ N − D − 1, then the kth lexicographic triangulation Lexk Q
of Q in the given vertex order is the triangulation obtained by pushing the first k − 1
vertices in the given order and then pulling the kth vertex. That is to say, pushing v1
creates a subdivision of Q = Q0 that has Q1 as its only nonsimplex cell.
Subsequently, the cell Q1 gets replaced by a pushing subdivision of Q1 with respect to v2,
and so on. Finally, pulling vk+1 in Qk completes the triangulation. The following
lemma asserts that the above procedure yields a regular subdivision by giving a
description in the spirit of Proposition 3.4.
Lemma 3.5 ([9, Example 2], [
13
]) Let ε > 0 and ω = (ω1, ω2, . . . , ωN−D−1, 0,
. . . , 0) ∈ RN be a set of normalized heights satisfying |ωi+1| ≤ ε|ωi | for all 1 ≤
i ≤ N − D − 2. If ε > 0 is sufficiently small, then Gω is a Gale transform encoding
Lexk Q for
k = min{i : ωi < 0} ∪ {n − d − 1}.
Definition 3.6 We call the polytope Lk(Q) = Q˜ ω corresponding to Gω the kth
lexicographic pyramid of Q.
According to the remarks following Proposition 3.4, Lk(Q) carries the combina
torics of Q as well as that of Lexk Q. So every facet of Lk(Q) is either a pyramid
over a facet of Q or a cell of Lexk Q.
We are now in a position to determine the combinatorics of NCPn,d (Q). To be
more precise, we determine the local combinatorial structure, i.e., for any given
vertex, we describe the set of facets that contain it. The construction of a neighborly
cubical polytope depended on an ordering of the vertices of Q, which we fix for the
following theorem.
Theorem 3.7 Let C = Cn(Q) be the deformed cube with respect to Q. Further, let
v ∈ C be an arbitrary vertex with eq v given by σ ∈ {+, −}n. Then the vertex figure
of πd (v) in NCPn,d (Q) is isomorphic to Lp(Q) for
p = min i ∈ [n] : σi = + ∪ {n − d − 1}.
In particular, the (d − 1)-faces of C containing v that are preserved by projection are
in one-to-one correspondence to the facets of Lp(Q).
Proof After a suitable base transformation of (2) by means of column operations, the
first n − d columns of Aeq v can be assumed to be of the form
with
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
ωi = (−1)i εi
i
j=1
σj .
By Lemma 3.5, this is a Gale transform of Lk(Q) with k = p.
Any generic projection of polytopes π : P → P = π(P ) induces a (contravariant)
order and rank preserving map π # : F ∂P → F ∂P .
The face poset of ∂NCPn,d (Q)/u, the boundary complex of the vertex figure of
u = πd (v) in NCPn,d (Q), is isomorphic to π #(F ∂NCPn,d (Q)≥u), the image of the
principal filter of u. By the Projection Lemma, the image coincides with the
embedding of Lp(Q) into the vertex figure F ∂Cn(Q)≥v .
Theorem 3.7 implies that the quotient NCPn,d (Q)/e with respect to certain edges
is isomorphic to Q. This implies the following result, which is already implicit in [
2
];
see also [6, Theorem 3.9].
Corollary 3.8 Nonisomorphic neighborly simplicial (d − 2)-polytopes Q and Q
yield nonisomorphic neighborly cubical polytopes NCPn,d (Q) and NCPn,d (Q ).
Moreover, there are at least as many different combinatorial types of d -dimensional
neighborly cubical polytopes as there are neighborly simplicial (d − 2)-polytopes on
n − 1 vertices.
The number of combinatorial types of neighborly simplicial polytopes is huge:
see [
15
].
3.2 Neighborly Cubical Polytopes from Cyclic Polytopes
In this section we (re)construct the neighborly cubical polytopes of Joswig and
Ziegler [
7
]. This specializes the discussion in the previous section to the case of Q a
cyclic polytope in the standard vertex ordering. By a thorough analysis of the
lexicographic triangulations of cyclic polytopes, we recover the “cubical Gale’s evenness
criterion” of [
7
]. For a treatment of cyclic polytopes and their triangulations beyond
our needs, we refer the reader to [
3, 16
].
The degree D moment curve is given by t → γ (t ) = (t, t 2, . . . , t D) ∈ RD . For
given pairwise distinct values t1, t2, . . . , tN ∈ R with N ≥ D + 1, the convex hull
of the corresponding points on the moment curve CycD(t1, . . . , tN ) = conv {γ (ti ) :
i ∈ [N ]} is a convex D-dimensional polytope. A fundamental consequence of the
theorem below is that the combinatorial type of CycD(t1, . . . , tN ) is independent of
the actual values ti . Therefore, we work with CycD(N ) := Cycd (1, 2, . . . , N ), the
D-dimensional cyclic polytope on N vertices in standard order. For the sake of
notational convenience later on, we describe its faces in terms of characteristic vectors of
cofaces: A vector α ∈ {0, 1}N names a coface of CycD(N ) iff conv {γ (i) : αi = 0} is
a face of CycD(N ). We also extend the notion of “co-” to subdivisions and, therefore,
speak freely about cocells.
Let α ∈ {0, 1}N be such that #{j < i : αj = 0} has the same parity for every i ∈ [N ]
with αi = 1. Then α is called even or odd according to this parity.
Theorem 3.9 (Gale’s Evenness Criterion [5, Sect. 4.7], [16, Theorem 0.7], [3,
Theorem 6.2.6]) A vector α ∈ {0, 1}N names a cofacet of CycD(N ) if and only if α has
exactly D zero entries and is either even or odd.
As a byproduct, we get that cyclic polytopes are:
• simplicial, since all facets have exactly D vertices;
• in general position, since every subpolytope is again cyclic;
• neighborly, since every α ∈ {0, 1}N with ≤ D2 zeros can be made to meet the
above conditions by changing entries 1 → 0.
From a geometric point of view, the odd and even (co)facets correspond to the
upper and lower facets of CycD(N ) with respect to the last coordinate. This dichotomy
among the facets allows for an explicit characterization of the (simplicial) cells of a
pushing/pulling subdivision of CycD(N ) with respect to the first vertex. Moreover,
since every vertex-induced subpolytope of CycD(N ) is again cyclic, from this we will
derive a complete description of the lexicographic triangulations of cyclic polytopes
with vertices in standard order.
To prepare for the precise statement, let Q = CycD(N ) = conv {vi = γd (i) : i ∈
[N ]} and Q = conv {v2, . . . , vN } =∼ CycD(N − 1) the subpolytope on all vertices
except the first. Let Γ be the subdivision of Q obtained by pulling or pushing v1.
Any cell in Γ that contains v1 is a D-simplex and, therefore, let α ∈ {0, 1}N be a
cocell with D + 1 zero entries and α1 = 0. Indeed, any such cell is a pyramid over
a facet of Q , and thus α is of the form α = (0, α ), and α adheres to the Gale’s
evenness criterion. The cocell α is part of a pushing or a pulling subdivision of Q if
and only if α is or is not a cofacet of Q. Clearly, the first gap in α is even, and, hence,
the parity of the gaps of α concludes the characterization.
Lemma 3.10 Let Q = CycD(N ) be a cyclic polytope, and let Lk(Q) be a
lexicographic pyramid of Q. Let α ∈ {0, 1}N+1 with D + 1 zero entries, and let p = min{i :
αi = 0}. Thus α is of the form
α = (1, 1, . . . , 1, 0, α ).
p−1
Then α is a cofacet of Lk(Q) if and only if one of the following conditions is
satisfied:
(i) 1 = p, and α is a cofacet of Cycd (n);
(ii) 1 < p < k, and α is even;
(iii) p = k, and α is odd.
Proof Every facet containing the 0th vertex is a pyramid over a facet of Q, and every
incident facet is of the form α = (0, α ) with α a cofacet of Q.
If 2 ≤ p < k, then α names a cocell of the pushing subdivision of Qp−1 =
conv {vp, . . . , vN } with respect to vp and containing vp. This, however, is the case
if and only if α is an even cofacet of Qp. The case p = k follows from similar
considerations.
Setting N = n − 1 and D = d − 2 and combining the above description with
Theorem 3.7, we obtain the following result of Joswig and Ziegler.
Theorem 3.11 (Cubical Gale’s Evenness Condition [
7
]) Let F be a (d − 1)-face
of the deformed cube C = Cn(Q) with Q := Cycd−2(n − 1). Let eq F be given by
α ∈ {+, −, 0}n, and let p ≥ 1 be the smallest index such that αp = 0. The face F
projects to a facet of NCPn,d (Q) if and only if α is of the form
α = (−, −, · · · , −, σ, 0, α )
p−2
| p+1|, . . . , |αn|) ∈ {0, 1}n−p satisfying the ordinary Gale’s evenness
with |α | = ( α
condition and if, for p > 1, one of the following conditions holds:
(i) σ = −, and |α | is even, or
(ii) σ = +, and |α | is odd.
Proof Let v ∈ F ⊂ C be a vertex with equality set β = eq v and such that βp = +.
By Theorem 3.7, the vertex figure of πd (v) in NCPn,d (Q) is isomorphic to Lk(Q),
with k ∈ {p − 1, p}.
Thus F projects to a facet of NCPn,d (Q) if and only if |α| is a cofacet of Lk(Q).
The result now follows from Lemma 3.10 by noting that k = p − 1 iff σ = −.
4 Deformed Products of Polygons
The projected deformed products of polygons (PDPPs) are four-dimensional
polytopes. They were constructed in [
18
] because of their extremal f -vectors: For
these polytopes, the fatness parameter Φ(P ) := ff01++ff32−−1200 is large, getting
arbitrarily close to 9. This parameter, introduced in [
17
], is crucial for the f -vector theory of
4-polytopes. In [
18
] the f -vectors of the PDPPs were computed without having a
combinatorial characterization of the polytopes in reach.
However, the PDPPs are yet another instance of projections of deformed products,
so the theory developed here gives us a firm grip on their properties. In the following
we generalize the construction to higher dimensions and analyze its combinatorial
structure using the tools developed in this paper. In particular, a description of the
facets of the PDPPs appears for the first time.
To begin with, the following is a generalization of Theorem 3.2.
Theorem 4.1 Let m ≥ 4 be even. For every 2 ≤ d ≤ 2r , there is a d-polytope whose
( d2 − 1)-skeleton is combinatorially isomorphic to that of an r -fold product of
m-gons.
Let us remark that the proofs of the results in this section can be adapted to yield
the generalizations for products of even polygons with varying numbers of vertices
in each factor. However, the generalized results require more technical and notational
overhead. Therefore, we trade generality in for clarity and only give the uniform
versions of the results.
For m = 4, the r -fold product of quadrilaterals is actually a cube of dimension
n = 2r, and thus NCPn,d (Q) satisfies the claims made. In the inequality description
the quadrilaterals can be seen by pairing up the intervals indicated by the framed
submatrices below:
. . .
We wish to build on this special case and therefore consider the normals of such a
quad:
The polygons we are heading for arise as generalizations of the above quad. For
m ≥ 4 even, consider the vectors
a0 = (−1, 0),
ai = 1, ε mm−−22i
for i = 1, . . . , m − 1,
as shown below. For suitable b0, b1, . . . , bm−1 > 0,
describes a convex m-gon in the plane:
aiTx ≤ bi for i = 0, . . . , m − 1
For the finishing touch, we scale every even-indexed inequality by ε,
We arrange the scaled normals and right-hand sides into a matrix and vector,
respectively,
εa0 ⎞
a1 ⎟
εa2 ⎟⎟
... ⎟⎟
⎠
am−1
and
Using these special polygons, we set up a template for a deformed product of
polygons (DPP).
Definition 4.2 (DPP template) For m ≥ 4 even and 2r ≥ d ≥ 2, let G = {g1, . . . ,
gd−1} ⊂ R2r−d be an ordered collection of row vectors. We denote by P2r (G; m) the
deformed product of polygons with lhs inequality system
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
A
1
A
In the above inequality system, the framed blocks denote matrices of appropriate
sizes that contain the depicted block repeated row-wise m2 times. In particular,
1
Proof of Theorem 4.1 Let P = P2r (G; m) be the deformed product of m-gons
according to the DPP template (3) which is determined by a Gale transform
G =
Id−2
G
of a neighborly simplicial (d − 2)-polytope Q with 2r − 1 ordered vertices in
general position. Equipped with a suitable right-hand side, the polytope P is an
iter(4)
ated rank 2 deformed product of polygons and thus combinatorially equivalent to the
r -fold product of an m-gon.
Now for an arbitrary vertex v of P , the matrix Aeq v is of the form
ai3
ai4
. . . . . .
1
.
The equality set of a vertex v is formed by two cyclically adjacent facets from each
polygon in the product. This means, in particular, that from each polygon there is an
even and an odd facet present in eq v. Every such pair is of the form
The absolute values of the diagonal entries are bounded by ε, while |ai +1 | < ε2.
Thus, provided that ε is sufficiently small, the rows of Aeq v below the
horizontal bar in (4) constitute a Gale transform of a polytope combinatorially equivalent
to Q.
In analogy to the cubical case, we write P2r (Q; m) for the deformed product of
m-gons with respect to the polytope Q with ordered vertices.
Definition 4.3 The proof of Theorem 4.1 yields a family of projected products of
polygons (PDPPs) as the image PDPP2r,d (Q; m) := πd (P2r (Q; m)).
En route to a facial description of PDPP2r,d (Q; m), let us pause to introduce a con
venient notation for handling products of even polygons combinatorially that bears
certain similarities with that of 2r -cubes, i.e., products of quadrilaterals. For the even
polygons above, we label the edge with outer normal ai by (i, ∗) if i is even and by
(∗, i) otherwise:
Every vertex is incident to an even edge (2i, ∗) and an odd edge (∗, 2i ± 1) and is la
beled by (2i, 2i ± 1). Finally, the polygon itself gets the label (∗, ∗) as the intersection
of no edges.
Summing up, the nonempty faces of an even m-gon are given by
∪ (∗, ∗)
(even edges)
(odd edges)
(vertices)
(polygon)
with inclusion given by the order relation induced by i ≺ ∗ for i ∈ {0, . . . , m − 1}.
Admittedly, this is neither the most natural nor the most efficient way to encode
a polygon combinatorially. However, the following remarks make up for this
unusual description. Similar to the description of 2r -cubes, the dimension of a face
(α0, α1) ∈ Pm is the number of ∗-entries. This carries over to products of
m-gons, i.e., there is an order-preserving bijection between the nonempty faces of an
r -fold product of m-gons and the r -fold direct product (Pm)r with rank function
dim α = #{i : αi = ∗} for α ∈ (Pm)r . Notably most of the results (and proofs) from
Sect. 3 carry over to this setting, with only minor modifications.
The key to obtaining a combinatorial description of PDPP2r,d (Q; m) is that, for
a vertex v of P2r (Q; m), the matrix (4) again encodes a lexicographic triangulation
of Q. In order to reduce this to the case of neighborly cubical polytopes, after a
suitable change of basis, the matrix Aeq v is of the form
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
The entries above the diagonal of ones remain to be of order ε. To determine the signs
of the entries, which will determine the lexicographic triangulation, let us investigate
the local change of the matrix under the change of basis.
In the above combinatorial model for even m-gons, the vertex v is identified with
a vector α = (α1, α2; α3, . . . ; α2r−1, α2r ) ∈ (Pm)r , which corresponds to eq v as
indicated. The following table, which is easily established given the coordinates of the
normals, summarizes the possible sign patterns in terms of α:
(αi , αi+1)
We use the last row, which gathers sign patterns from the diagonal, to define the map
Φ : (α1, α2) ∈ Pm : α vertex
→ {+, −, 0}2
with Φ(α1, α2) := (σ1, σ2) according to the table. Since the face lattice of a convex
polytope is atomic, it is easy to see from the definition that Φ : Pm → {+, −, 0}2
extends to an order- and rank-preserving map from the face poset of an even m-gon
to that of a 2-cube. The map can be thought of as a folding map:
The induced map Φ : (Pm)r → {+, −, 0}2r maps faces of P2r (k; m) that are
strictly preserved under πd to surviving faces of C2r (Q). Phrased differently, the
following diagram commutes on the level of faces:
Pn,r (Q)
⏐⏐ πd
Φ
−−−−−−−−−−−→
C2r (Q)
⏐⏐ πd
Φ
PDPP2r,d (Q; m) −−−−−−−−−−−→ NCPd (Q).
Proposition 4.4 Let n = 2r , and let P = Pn(Q; m) and C = Cn(Q) be the
deformed cube and the product of m-gons of dimension n = 2r with respect to a
neighborly simplicial (d − 2)-polytope Q on n − 1 ordered vertices. Let v ∈ P be
a vertex with eq v represented by α ∈ (Pm)r , and let u ∈ C be the vertex
corresponding to Φ(α) ∈ {+, −}n. Then Φ induces an isomorphism of the vertex figures
PDPPn,d (Q; m)/πd (v) and NCPn(Q)/πd (u).
Proof As consistent with the main theme in this article, consider the first n − d =
2r − d coordinates of the inequalities from both P and C that are tight at v and u,
respectively.
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛ ai1
1
a˜i2
. . .
Av(P )
g˜d−1
⎛ σi1 ε
1
σi2 ε
. . .
In both matrices, the entries on the secondary diagonal are arbitrary small, and the
map Φ assures that corresponding entries have equal sign. By Lemma 3.5, both
Av(P ) and Au(C) are Gale transforms that encode the same lexicographic
pyramid Lk(Q). The result now follows by observing that a face β α of P is strictly
preserved if and only if |β| is a coface of Lk(Q) and |Φ(β)| = |β|.
This proposition makes way for the combinatorics of the projected deformed
products associated with arbitrary neighborly simplicial polytopes.
Theorem 4.5 (Combinatorial description of the PDPPs) Let P = P2r (Q; m) be a
deformed product of m-gons with respect to Q, and let v ∈ P be an arbitrary vertex with
eq v = α ∈ (Pm)r . Then the vertex figure of πd (v) in PDPP2r,d (Q; m) is isomorphic
to Lp(Q) for
p = min i ∈ [2r] : Φ(α)i = − ∪ {2r − d − 1}.
In particular, the (d − 1)-faces of P containing v that are preserved by projection
are in one-to-one correspondence to the facets of Lp(Q).
As for the neighborly cubical polytopes, via Shemer’s work [
15
] this result implies
a great richness of combinatorial types for the projected products of polygons. In the
special case where Q is a cyclic polytope with vertices in standard order, we get a
very explicit Gale’s evenness-type criterion for the projected products of polygons.
Corollary 4.6 (Combinatorial description of the standard PDPPs) Let F ⊂ P =
P2r (Q; m) be a (d − 1)-face with Q = Cycd−2(2r − 1), and let β ∈ (Pm)r
correspond to eq F . Then F projects to a facet of PDPP2r,d (Q; m) if and only if Φ(β)
satisfies the cubical Gale’s evenness criterion.
Acknowledgements The first author would like to thank Andreas Paffenholz, Thilo Rörig, Jakob
Uszkoreit, Arnold Waßmer, and Axel Werner for “actively listening” and Vanessa Kääb for more. Both authors
gratefully acknowledge support by the German Science Foundation DFG via the Research Training Group
“Methods for Discrete Structures” and a Leibniz grant.
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