Unconditional reflexive polytopes

A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gr\"obner bases for unconditional reflexive polytopes coming from partially ordered sets


INTRODUCTION
A d-dimensional convex lattice polytope P ⊂ R d is called reflexive if its polar dual P * is again a lattice polytope. Reflexive polytopes were introduced by Batyrev [Bat94] in the context of mirror symmetry as a reflexive polytope and its dual give rise to a mirror-dual pair of Calabi-Yau manifolds (c.f. [Cox15]). As thus, the results of Batyrev, and the subsequent connection with string theory, have stimulated interest in the classification of reflexive polytopes both among mathematical and theoretical physics communities. As a consequence of a well-known result of Lagarias and Ziegler [LZ91], there are only finitely many reflexive polytopes in each dimension, up to unimodular equivalence. In two dimensions, it is a straightforward exercise to verify that there are precisely 16 reflexive polygons, as depicted in Figure 1. While still finite, there are significantly more reflexive polytopes in higher dimensions. Kreuzer and Skarke [KS98,KS00] have completely classified reflexive polytopes in dimensions 3 and 4, noting that there are exactly 4319 reflexive polytopes in dimension 3 and 473800776 reflexive polytopes in dimension 4. The number of reflexive polytopes in dimension 5 is not known.
In recent years, there have been a number of results characterizing reflexive polytopes in known classes of polytopes coming from combinatorics or optimization; see, for example, [BHS09,Tag10,Ohs14,HMT15,CFS17]. The purpose of this paper is to study a class of reflexive polytopes motivated by convex geometry and relate it to combinatorics. A convex body K ⊂ R d is unconditional if p ∈ K if and only if σp := (σ 1 p 1 , σ 2 p 2 , . . . , σ d p d ) ∈ K for all σ ∈ {−1, +1} d . Unconditional convex bodies, for example, arise as unit balls in the theory of Banach spaces with a 1-unconditional basis. They constitute a restricted yet surprisingly interesting class of convex bodies for which a number of claims have been verified; cf. [BGVV14]. For example, we mention that the Mahler conjecture is known to hold for unconditional convex bodies; see Section 3. In this paper, we investigate unconditional lattice polytopes and their relation to anti-blocking polytopes from combinatorial optimization. In particular, we completely characterize unconditional reflexive polytopes.
The structure of this paper is as follows. In Section 2, we briefly review notions and results from discrete geometry and Ehrhart theory.  [Hof18].
In Section 3, we introduce and study unconditional and, more generally, locally anti-blocking polytopes. The main result is Theorem 3.2 that relates regular, unimodular, and flag triangulations to the associated anti-blocking polytopes.
In Section 4, we associate an unconditional lattice polytope UP G to every finite graph G. We show in Theorems 4.6 and 4.9 that an unconditional polytope P is reflexive if and only if P = UP G for some unique perfect graph G. This also implies that unconditional reflexive polytopes have regular, unimodular triangulations.
Section 5 is devoted to a particular family of unconditional reflexive polytopes and is of independent interest: We show that the type-B Birkhoff polytope or signed Birkhoff polytope BB(n), that is, the convex hull of signed permutation matrices, is an unconditional reflexive polytope. We compute normalized volumes and h * -vectors of BB(n) and its dual C(n) = BB(n) * for small values of n.
The usual Birkhoff polytope and the Gardner polytope of [FHSS] appear as faces of BB(n) and C(n), respectively. These two polytopes form a Gale-dual pair in the sense of [FHSS]. In Section 6, we give a general construction for compressed Gale-dual pairs coming from CIS graphs.
In Section 7, we investigate unconditional polytopes associated to comparability graphs of posets. In particular, we explicitly describe a quadratic square-free Gröbner basis for the corresponding toric ideal.
We close with open questions and future directions in Section 8.
Acknowledgements. The first two authors would like to thank Matthias Beck, Benjamin Braun, and Jan Hofmann for helpful comments and suggestions for this work. Furthermore, Figure 2 was created by Benjamin Schröter. Additionally, the authors thank Takayuki

BACKGROUND
In this section, we provide a brief introduction to polytopes and Ehrhart theory. For additional background and details, we refer the reader to the excellent books [BR15,Zie95]. A polytope in R d is the inclusion-minimal convex set P = conv(v 1 , . . . , v n ) containing a given collection of points v 1 , . . . , v n ∈ R d . If v 1 , . . . , v n ∈ Z d , then P is called a lattice polytope. The unique inclusionminimal set V ⊆ P such that P = conv(V ) is called the vertex set and is denoted by V (P ). By the Minkowski-Weyl theorem, polytopes are precisely the bounded sets of the form is a facet and the inequality is facet-defining.
The dimension of a polytope P is defined to be the dimension of its affine span. A d-dimensional polytope has at least d + 1 vertices and a d-polytope with exactly d + 1 many vertices is called a where vol is the Euclidean volume. For lattice polytopes P ⊂ R d , we define the normalized volume Vol(P ) := d! vol(P ). So unimodular simplices are the lattice polytopes with normalized volume 1. We say that two lattice polytopes P, P ′ ⊂ R d are unimodularly equivalent if P ′ = T (P ) for some transformation T (x) = W x + v with W ∈ SL d (Z) and and v ∈ Z d . In particular, any two unimodular simplices are unimodularly equivalent.
Given a lattice d-polytope P and t ∈ Z ≥1 , let tP := {t · x : x ∈ P } be the t th dilate of P . The lattice-point enumeration function is called the Ehrhart polynomial. By a famous result of Ehrhart [Ehr62, Thm. 1], this function agrees with a polynomial in the variable t of degree d with leading coefficient vol(P ).
This also implies that the formal generating function is a rational function with denominator (1 − z) d+1 and that the degree of the numerator is at most d (see, e.g., [BR15, Lem 3.9]). We call the numerator the h * -polynomial of P . The coefficient vector h * P = (h * 0 , h * 1 , . . . , h * d ) ∈ Z d+1 is called the h * -vector of P . One should note that the Ehrhart polynomial is invariant under unimodular transformations.
Theorem 2.1 ( [Sta80,Sta93]). Let P ⊆ Q be a lattice polytopes. Then The h * -vector encodes a lot of information about the underlying polytope. This is nicely illustrated in the case of reflexive polytopes. For a d-polytope P ⊂ R d with 0 in the interior, we define the (polar) dual polytope Definition 2.2. Let P ⊂ R d be a d-dimensional lattice polytope that contains the origin in its interior. We say that P is reflexive if P * is also a lattice polytope. Equivalently, P is reflexive if it has a description of the form where A is an integral matrix.
Quite surprisingly, reflexivity can be completely characterized by enumerative data of the h *vector.
The reflexivity property is also deeply related to commutative algebra. A polytope P is reflexive if the canonical module of the associated graded algebra k[P ] is (up to a shift in grading) isomorphic to k[P ] and its minimal generator has degree 1. If one allows the unique minimal generator to have arbitrary degree, one arrives at the notion of Gorenstein rings, for details we refer to [BG09, Sec 6.C]. We say that P is Gorenstein if there exists a c ∈ Z ≥1 such that cP is unimodularly equivalent to a reflexive polytope. This is equivalent to saying that k[P ] is Gorenstein. The dilation factor c is often called the codegree. In particular, reflexive polytopes are Gorenstein of codegree 1. By combining results of Stanley [Sta78] and De Negri-Hibi [DNH97], we have a characterization of the Gorenstein property in terms of the h * -vector. Namely, P is Gorenstein if and only if h * i = h * d−c+1−i for all i. Aside from examining algebraic properties of lattice polytopes, one can also investigate discrete geometric properties. Every lattice polytope admits a subdivision into lattice simplices. Even more, one can guarantee that every lattice point contained in a polytope corresponds to a vertex of such a subdivision. However, one cannot guarantee the existence of a subdivision where all simplices are unimodular when the dimension is greater than 2. This leads us to our next definition: Definition 2.4. A triangulation T of a lattice d-dimensional polytope P with vertices in V is a collection of lattice d-dimensional simplices with vertices in V covering P and such that any two simplices meet in a common face. We call T unimodular if all simplices are unimodular.
A triangulation T is regular if there is a convex, piecewise-linear function ω : P → R whose domains of linearity are exactly the simplices in T . Such a function is completely described by assigning values ω(v) for v ∈ V .
A triangulation is flag if the inclusion-minimal sets of vertices not forming a face in T are all of cardinality 2.
Given a lattice polytope P , a pulling triangulation is a triangulation obtained by a sequence of pulling refinements. Given v ∈ P ∩ Z d and a lattice subdivision S, pull v P is the refined lattice subdivision induced by replacing every face F ∈ S such that v ∈ F with the pyramids conv(v, F ′ ), for each face F ′ of F that does not contain v. Such refinements preserve regularity and thus a triangulation constructed by a sequence of pulling refinements is a regular triangulation. The reader should consult [DLRS10,HPPS14] for more details.
A special class of polytopes which possess regular, unimodular triangulations are compressed polytopes. A polytope P is compressed if every pulling triangulation is unimodular [Sta80]. In the interest of providing a useful characterization of compressed polytopes, we must define the notion of width of a facet. Let P ⊂ R d be a d-dimensional lattice polytope and F i = P ∩ {x : a i , x = b i } a facet. We assume that a i is primitive, that is, its coordinates are coprime. The width of F i is (1) P is compressed; (2) P has width one with respect to all its facets; (3) P is unimodularly equivalent to the intersection of a unit cube with an affine space.
Definition 2.6. A lattice polytope P has the integer decomposition property (IDP) if for any positive integer t and for all One should note that if P has a unimodular triangulation, then P has the IDP. However, there are examples of polytopes which have the IDP, yet do not even admit a unimodular cover, that is, a covering of P by unimodular simplices, see [BG99,Sec. 3]. A more complete hierarchy of covering properties can be found in [HPPS14]. We Unimodality appears frequently in combinatorial settings and it often hints at a deeper underlying algebraic structure, see [AHK18,Bre94,Sta89]. One famous instance is given by Gorenstein polytopes that admit a regular, unimodular triangulation.
The following conjecture is commonly attributed to Ohsugi and Hibi [OH06]: Conjecture 2.8. If P is Gorenstein and has the IDP, then h * P is unimodal.

UNCONDITIONAL AND ANTI-BLOCKING POLYTOPES
So, P is a polytope that is symmetric with respect to all coordinate hyperplanes. It is apparent that P can be recovered from its restriction to the first orthant, which we denote by P + = P ∩R d + . The polytope P + has the property that for any q ∈ P + and p ∈ R d with 0 ≤ p i ≤ q i for all i, it holds that p ∈ P + . Polytopes in R d + with this property are called anti-blocking polytopes. Anti-blocking polytopes were studied and named by Fulkerson [Ful71,Ful72] in the context of combinatorial optimization, but they are also known as convex corners or down-closed polytopes; see, for example, [BB00].
Let us also write p = (|p 1 |, |p 2 |, . . . , |p d |). Given an anti-blocking polytope Q ⊂ R d + it is straightforward to verify that UQ := {p ∈ R d : p ∈ Q} is an unconditional convex body. Every full-dimensional anti-blocking polytope has an irredundant inequality description of the form as the smallest anti-blocking polytope containing c 1 , . . . , c r ∈ R d + . Conversely, if we let V ↓ (Q) = {v 1 , . . . , v m } be the vertices of Q that are maximal with respect to the componentwise order, then Q = {v 1 , . . . , v r } ↓ . We record the consequences for the unconditional polytopes.
Proposition 3.1. Let P ⊂ R d + be an anti-blocking polytope given by (1). Then an irredundant inequality description of UP is given by the distinct Our first result relates properties of subdivisions of anti-blocking polytopes to that of the associated unconditional polytopes. The 2 d orthants in R d are denoted by R d σ := σR d + for σ ∈ {−1, +1} d . Theorem 3.2. Let P ⊂ R d + be an anti-blocking polytope with triangulation T . Then Proof. It is clear that UT is a triangulation of UP and statement (i) is obvious. If U is a collection of vertices of T not contained in R d σ for any σ, then there are u + , u − ∈ U that are not contained in the same orthant. Hence if T is flag, then UT is flag, which proves (iii).
To show (ii), assume that T is regular. Let ω : V (P ) → R the corresponding heights. We extend For ǫ = 0 it is easy to see that the heights induce a subdivision of P into σP for σ ∈ {−1, +1} d . For ǫ > 0 sufficiently small, the heights ω ′ then induce the triangulation σT on σP .
Let us call a polytope P ⊂ R d locally anti-blocking if (σP ) ∩ R d + is an anti-blocking polytope for every σ ∈ {−1, +1} d . Unconditional polytopes are clearly locally anti-blocking. It follows from [CFS17, Lemma 3.12] that for any two anti-blocking polytopes P 1 , P 2 ⊆ R d + , the polytopes P 1 + (−P 2 ) and are locally anti-blocking. Locally anti-blocking polytopes are studied in depth in [AASS19]. The following is a simple but important observation.

Lemma 3.3. Let P ⊂ R d be a locally anti-blocking lattice polytope. Then P is reflexive if and only if
Since Q is full-dimensional, we have that the standard basis vectors e i = (0, . . . , 0, 1, 0, . . . , 0) are contained in Q and thus (a i ) j ∈ {0, b i }. Since we assume a i primitive, it follows that b i = 1.
Hence, from the definition of reflexive polytopes and Proposition 3.1 we infer that P is reflexive if and only if P σ is of the form {x : A σ x ≤ 1} for some integer matrix A σ for every σ ∈ {−1, +1} d .
Theorem 3.4. If P is a reflexive and locally anti-blocking polytope, then P has a regular and unimodular triangulation. In particular, h * (P ) is unimodal.
Proof. By Lemma 3.3, every pulling triangulation of P σ = P ∩ R d σ for σ ∈ {0, 1} d is a unimodular triangulation. Let U = P ∩ Z d and choose an ordering of the points in U such that u comes before v if the support of u is contained in the support of v. This gives a consistent pulling order of the vertices of each P σ . The same argument as in the proof of Theorem 3.2, then shows that the regular subdivision of P into the polytopes P σ can be refined to a regular and unimodular triangulation T . The unimodality of h * (P ) now follows from Theorem 2.7.
Remark 3.5. The techniques of this section can be extended to the following class of polytopes. We say that a polytope P ⊂ R d has the orthant-lattice property (OLP) if the restriction P σ := P ∩R d σ is a (possibly empty) lattice polytope. If P is reflexive, then P σ is full-dimensional for every σ. Now, if every P σ has a unimodular cover, then so does P and hence is IDP.
Then some conditions that imply the existence of a unimodular cover include: (1) P σ is compressed; (2) A σ is a totally unimodular matrix; (3) A σ consists of of rows which are B d roots; (4) P σ is the product of unimodular simplices; (5) There exists a projection π : R d → R d−1 such that π(P σ ) has a regular, unimodular triangulation T such that the pullback subdivision π * (T ) is lattice. We refer to [HPPS14] for background and details.
An example of such a polytope is This is a reflexive OLP polytope. The restriction to R 3 + is which is not an anti-blocking polytope.
The Mahler conjecture in convex geometry states that every centrally-symmetric convex body The Mahler conjecture has been verified only in small dimensions and for special classes of convex bodies. In particular, Saint-Raymond [SR80] proved the following beautiful inequality. The characterization of the equality case is independently due to Meyer [Mey86] and Reisner [Rei87]. This inequality directly implies the Mahler conjecture for unconditional convex polytopes, that we record for the normalized volume.
Corollary 3.7. Let P ⊂ R d be an unconditional reflexive polytope. Then with equality if and only if P or P * is the cube [−1, 1] d .

UNCONDITIONAL REFLEXIVE POLYTOPES AND PERFECT GRAPHS
is a simplicial complex, i.e., a nonempty set system closed under taking subsets, then is an anti-blocking 0/1-polytope and every anti-blocking polytope with vertices in {0, 1} d arises that way. A prominent class of anti-blocking 0/1-polytopes arises from graphs. Given is a stable set (or independent set) of G if uv ∈ E for any u, v ∈ S. The stable set polytope of G is Stable set polytopes played an important role in the proof of the weak perfect graph conjecture [Lov72]. Lovász gave the following geometric characterization of perfect graphs. For a set C ⊆ [d] and x ∈ R d , we write x(C) = i∈C x i .

Theorem 4.1. A graph G = ([d], E) is perfect if and only if
For an anti-blocking polytope P ⊂ R d + define the anti-blocking dual A(P ) := {y ∈ R d + : y, x ≤ 1 for all x ∈ P } . The polar (UP ) * is again unconditional and it follows that From Theorem 4.1 one then deduces for a perfect graph G that

Corollary 4.3 (Weak perfect graph theorem). A graph G is perfect if and only if G is perfect.
We note that in particular if G is perfect, then P G is compressed. Let us remark that Theorem 4.1 also allows us to characterize the Gorenstein stable set polytopes. For comparability graphs of posets (see Section 7) this was noted by Hibi [Hib87]. A graph G is called well-covered if every inclusion-maximal stable set has the same size. It is called cowell-covered if G is well-covered. Theorem 4.6. Let P ⊂ R d be a locally anti-blocking lattice polytope. Then P is reflexive if and only if for every σ ∈ {−1, +1} d there is a perfect graph G σ such that P σ = P Gσ .
In particular, P is an unconditional reflexive polytope if and only if P = UP G for some perfect graph G. For G 1 = G 2 = K d the complete graph on d vertices, the polytope P G 1 + (−P G 2 ) is the Legendre polytope studied by Hetyei et al. [Het09,EHR18].
Using NORMALIZ [BIR + ] and the Kreuzer-Skarke database for reflexive polytopes [KS98, KS00], we were able to verify that 72 of the 3-dimensional reflexive polytopes and at least 407 of the 4dimensional reflexive polytopes with at most 12 vertices are locally anti-blocking. Unfortunately, our computational resources were too limited to test most of the 4-dimensional polytopes. However, there are only 11 4-dimensional unconditional reflexive polytopes (by virtue of Theorem 4.9).
If G, G ′ are perfect graphs, then G ⊎ G ′ as well as its bipartite sum G ⊲⊳ G ′ = G ⊎ G ′ are perfect. On the level of unconditional polytopes we note that These observations give us the class of Hanner polytopes which are important in relation to the 3 d -conjecture; see [SWZ09]. Let us briefly note that Theorem 4.6 also yields bounds on the entries of the h * -vector. Recall that h * i (C d ) for the cube C d = [−1, +1] d is given by the type-B Eulerian number B(n, i) = i j=1 (−1) k−i n j−i 2 j−1 n−1 that counts signed permutations with i descents (see also Section 5).
Corollary 4.8. Let P ⊂ R d be an unconditional reflexive polytope. Then Proof. It follows from Theorem 4.6 that every reflexive and unconditional P satisfies C * d ⊆ P ⊆ C d , where C d = [−1, 1] d . By Theorem 2.1, the entries of the h * -vector are monotone with respect to inclusion.
We close the section by showing that distinct perfect graphs yield distinct unconditional reflexive polytopes. Proof. Assume that T (UP G ) = UP H for some T (x) = W x + t with t ∈ Z d and W ∈ SL d (Z). Since the origin is the only interior lattice point of both polytopes, we infer that t = 0. Let W = (w 1 , . . . , w d ). Thus, z ∈ Z d is a lattice point in UP H if and only if there is a stable set S and σ ∈ {−1, +1} S such that On the one hand, this implies that w i and w j have disjoint supports whenever i, j ∈ S and i = j. Indeed, if the supports of w i and w j are not disjoint, then σ i w i + σ j w j has a coordinate > 1 for some choice of σ i , σ j ∈ {−1, +1}, which contradicts the fact that UP H ⊆ [−1, 1] d .
On the other hand, for any h ∈ [d], the point e h is contained in UP H . Hence, there is a stable set S and σ ∈ {−1, +1} S such that (3) holds for z = e h . Since the supports of the vectors indexed by S are disjoint, this means that S = {i} and e h = σ i w i . We conclude that W is a signed permutation matrix and G ∼ = H.
We can conclude that number of unconditional reflexive polytopes in R d up to unimodular equivalence is precisely the number of unlabelled perfect graphs on d vertices. This number has  been computed up to d = 13 (see [Hou06,Sec.5] and A052431 of [Slo19]). We show the sequence in Table 1.

THE TYPE-B BIRKHOFF POLYTOPE
Recall that the Birkhoff polytope B(n) is defined as the convex hull of all n × n permutation matrices or equivalently as the set of all doubly stochastic matrices, that is, nonnegative matrices M with row and column sums equal to 1, by work of Birkhoff [Bir46] and, independently, von Neumann [vN53]. This polytope has been studied quite extensively and is known to have many properties of interest (see, e.g., [Ath05,BR97,BP03,CM09,Dav15,DLLY09,Paf15]). Of particular interest to our purposes, it is known to be Gorenstein, to be compressed [Sta80], and to be h * -unimodal [Ath05]. In this section, we will introduce a type-B analogue of this polytope corresponding to signed permutation matrices and verify many similar properties already known for B(n).
The hyperoctahedral group is defined to by B n := Z/2Z≀S n , which is the Coxeter group of type-B (or type-C). Elements of this group can be thought of as permutations from S n expressed in one-line notation σ = σ 1 σ 2 · · · σ n , where we also associate a sign sgn(σ i ) to each σ i . To each signed permutation σ ∈ B n , we associate a matrix M σ defined as (M σ ) i,σ i = sgn(σ i ) and (M σ ) i,j = 0 otherwise. If every entry of σ is positive, then M σ is simply a permutation matrix. This leads to the following definition: Definition 5.1. The type-B Birkhoff polytope (or signed Birkhoff polytope) is That is, BB(n) is the convex hull of all n × n signed permutation matrices.
This polytope was previously studied in [MOSZ02], though the emphasis was not on Ehrharttheoretic questions. Since all points in the definition of BB(n) lie on a sphere, it follows that they are all vertices.
Proposition 5.2. For every σ ∈ B n , M σ is a vertex of BB(n).
It is clear that BB(n) is an unconditional lattice polytope in R d×d and we study it by restriction to the positive orthant. A simple way to view this as an anti-blocking polytope is via matching polytopes. Given a graph G = ([d], E), a matching is a set M ⊆ E such that e ∩ e ′ = ∅ for any two distinct e, e ′ ∈ M . The corresponding matching polytope is If G is a bipartite graph, then the matching polytope is easy to describe. For v ∈ [d] let δ(v) ⊆ E denote the edges incident to v. Sec. 8.11]). For bipartite graphs G the matching polytope is given by As a simple consequence, we get The polytope BB + (n) is the stable set polytope of L(K n,n ) = K n K n , the Cartesian product of complete graphs, which for obvious reasons is called a rook graph.
Since all vertices in K n,n have the same degree, it follows that all maximal cliques in K n K n have size n and from Proposition 4.5 we conclude the following.
Furthermore, we can deduce that BB(n) is an unconditional reflexive polytope by Theorem 4.6. For two matrices A, B ∈ R d×d we denote by A, B = tr(A t B) the Frobenius inner product. Also, for vectors u, v ∈ R d let us write u ⊗ v ∈ R d×d for the matrix with (u ⊗ v) ij = u i v j .
The inequality description of this polytope was previously obtained in [MOSZ02] using the notion of Birkhoff tensors. However, we ascertain this result by applying Proposition 3.1 and Theorem 5.4.
The dual C(n) := BB(n) * is the unconditional reflexive polytope associated with the graph K n K n . The corresponding anti-blocking polytope C + (n) = P Kn Kn also has the nice property that all cliques have the same size n and hence Proposition 4.5 applies.
By Theorem 3.4 and Proposition 4.4, we have the following unimodality results.
Corollary 5.9. For any n ∈ Z ≥1 , we have that h * BB(n) , h * BB + (n) , h * C(n) , and h * C + (n) are unimodal. Let us conclude this section with some enumerative data. The polytope BB(n) has 2 n n! vertices and n2 n+1 facets. In contrast, the vertices of BB + (n) are in bijection to partial permutations of [n]. Hence BB + (n) has n! n i=0 1 i! many vertices but only n 2 +2n facets. The polytope C + (n) has n2 n+1 − (n + 1) 2 many vertices and n 2 + n! facets. We used NORMALIZ [BIR + ] to compute the normalized volume and h * -vectors of these polytopes; see Tables 2, 3  volumes of these polytopes, our computational resources were quite quickly exhausted. Note that BB(3) and C(3) have precisely the same Ehrhart data and normalized volume and in fact it is straightforward to verify that BB(3) and C(3) are unimodularly equivalent.
Using Theorem 3.6 and Corollary 3.7, we get a lower bound on the volume of BB + (5) and BB(5), respectively. We get that are bounds on the number of simplices in an unimodular triangulation.

CIS GRAPHS AND COMPRESSED GALE-DUAL PAIRS OF POLYTOPES
The notion of Gale-dual pairs was introduced in [FHSS]. Given two polytopes P, Q ⊂ R d , we say that these polytopes are a Gale-dual pair if P = {x ∈ R d + : x, y = 1 for y ∈ V (Q)} and Q = {x ∈ R d + : x, y = 1 for y ∈ V (P )} .
n Vol(C(n)) h * C(n) The prime example of a Gale-dual pair of polytopes is the Birkhoff polytope B n , the convex hull of permutation matrices M τ , and the Gardner polytope G n , which is the polytope of all nonnegative matrices A ∈ R n×n + such that M τ , A = 1 for all permutation matrices M τ . Both polytopes are compressed Gorenstein lattice polytopes of codegree n. The question raised in [FHSS] was if there other Gale-dual pairs with (a subset of) these properties. In this section we briefly outline a construction for compressed Gale-dual pairs of polytopes.
Following [ABG18], we call G = ([d], E) a CIS graph if C ∩ S = ∅ for every inclusion-maximal clique C and inclusion-maximal stable set S. For brevity, we refer to those as maximal cliques and stable sets, respectively. For example, if B is a bipartite graph with perfect matching, then the line graph L(G) is CIS. Another class of examples is given by a theorem of Grillet [Gri69]. Let Π = ([d], ) be a partially ordered set. The comparability graph of Π is the simple graph Comparability graphs are known to be perfect. The bull graph is the graph vertices a, b, c, d, e and edges ab, bc, cd, de, bd. The wording in graph-theoretic terms is due to Berge; see [Zan95] for extensions. Proof. Note that every stable set meeting every maximal clique is necessarily a maximal stable set. Hence, it follows from Theorem 4.1 that P = {x ∈ R n + : x(C) = 1 for C maximal clique} . Since G is also a perfect CIS graph, the same holds for Q.
Note that both of the examples above are perfect and CIS graphs. This shows that compressed (lattice) Gale-dual pairs are not rare. Recall that a graph G is well-covered if every maximal stable set has the same size and G is co-well-covered if G is well-covered. Theorem 6.1 and its generalization in [Zan95] allow for the construction of perfect CIS graphs which are well-covered and co-well-covered (for example, by taking ordinal sums of antichains). Moreover, the recent paper [DHMV15] gives classes of examples of well-covered and co-well-covered CIS graphs. This is a potential source of compressed Gorenstein Gale-dual pairs but we were not able to identify the perfect graphs in these families.
Theorem 4.6 implies that if (F, G) is a Gale-dual pair of Proposition 6.2, then there is a (unconditional) reflexive polytope such that F ⊂ P and G ⊂ P * are dual faces. Question 6.3. Is it true that every Gale-dual pair (F, G) appears as dual faces of some reflexive polytope P ?

CHAIN POLYTOPES AND GRÖBNER BASES
Given a lattice polytope P ⊂ R d , the existence of regular triangulations, particularly those which are unimodular and flag, has direct applications to the associated toric ideal of P . In this section, we will discuss how the Gröbner basis of the toric ideal of anti-blocking polytope can be extended to the associated unconditional polytope. In particular, we provide an explicit description of the Gröbner bases of the unconditional polytopes arising from the special class of antiblocking polytopes called chain polytopes. We refer the reader to the wonderful books [CLO15] and [Stu96] for background on Gröbner bases and toric ideals.
Let Z := P ∩ Z d . The toric ideal associated to P is the ideal I P ⊂ C[x p : p ∈ Z] with generators where r 1 , . . . , r k , s 1 , . . . , s k ∈ Z such that r 1 + · · · + r k = s 1 + · · · + s k . If we denote the two multisets of points by R and S, we simply write x R − x S . A celebrated result of Sturmfels [Stu96,Thm. 8.3] states that the regular triangulations T of P (with vertices in Z) are in correspondence with (reduced) Gröbner bases of I P . As customary we write x R − x S to emphasize that x R is the leading term. In particular, if T is unimodular, then the leading terms of the associated Gröbner basis are square-free [Stu96, Cor. 8.9]. That is, R is an actual set. Given any lattice point p ∈ 2P there are unique p (1) , p (2) ∈ Z such that 2p = p (1) + p (2) and {p (1) , p (2) } is an edge in T . Let us call two points p, q ∈ R d separable if for some p i and q i have different signs for some i = 1, . . . , d. Together with Theorem 3.2, this yields the following description of a Gröbner basis for unconditional reflexive polytope.
Theorem 7.1. Let P ⊂ R d be an anti-blocking polytope with a regular, unimodular, flag triangulation and let UP be the associated unconditional polytope. Let x R i − x S i for i = 1, . . . , m be the Gröbner basis for I P . Then the following binomials give a Gröbner basis for I UP : x σR i − x σS i for i = 1, . . . , m and σ ∈ {−1, +1} d . Moreover, for any p, q ∈ UP ∩ Z d separable, let σ such that σ(p + q) = e ∈ 2P and let A prominent class of perfect graphs G for which regular, unimodular triangulations P G , as well as Gröbner bases for I P G , are well understood are comparability graphs of finite posets. Let Π = ([d], ) be a partially ordered set. with comparability graph G ≺ . The stable set polytopes associated to comparability graphs were studied by Stanley [Sta86] under the name chain polytopes and denoted by C(Π). The vertices of P G≺ are precisely points e A , where A is an antichain which is a collections of incomparable elements in Π. Let A(Π) denote the collection of antichains. The pulling triangulation of P G can be explicitly described (see Section 4.1 in [CFS17] for exposition and details). The corresponding (reverse lexicographic) Gröbner basis was described by Hibi [Hib87]. Following [CFS17], we define where min and max are taken with respect to the partial order . We call two antichains A, A ′ incomparable if there are a ∈ A and a ′ ∈ A ′ such that A ∪ {a ′ }, A ′ ∪ {a} ∈ A(Π). Equivalently, if max(A ∪ A ′ ) is a subset of neither A nor A ′ . To ease notation, we identify variables

Theorem 7.2. A Gröbner basis for I C(Π) is given by the binomials
for all incomparable antichains B, B ′ ∈ A(Π).
We define the unconditional chain polytope UC(Π) as the unconditional reflexive polytope associated to G ≺ . The lattice points in UC(Π) are uniquely described by Proof. In light of Theorems 7.1 and 7.2 , we only need to argue the second collection of binomials.
It follows from Theorem 7.2 that the edges of the unimodular (pulling) triangulation of C(Π) are of the form {D, D ′ } where D, D ′ are comparable antichains. That is, for every b ∈ D there is b ′ ∈ D ′ with b b ′ . For p ∈ 2C(Π), there are unique comparable D, D ′ ∈ A(Π) with 2p = e D + e D ′ . Set S := {i : p i ≥ 1} and T := {i : p i = 2}. Then it follows from the fact that every element in D ∪ D ′ is either a minimum or maximum that D = min(S) and D ′ = max(S) ∪ T . Hence if p = e C + e C ′ for arbitrary antichains C, C ′ , then D = C ⊓ C ′ and D ′ = C ⊔ C ′ . 8. CONCLUDING REMARKS 8.1. Birkhoff polytopes of other types. It is only natural to look at Birkhoff-type polytopes of other finite irreducible Coxeter groups. Since the type-B and the type-C Coxeter groups are equal, we get the same polytope. Recall that the type-D Coxeter group D n is the subgroup of B n with permutations with an even number of negatives. We can construct the type-D Birkhoff polytope, BD(n), to be the convex hull of signed permutation matrices with an even number of negative entries. As one may suspect from this construction, the omission of all lattice points in various orthants which occurs in BD(n) ensures that it cannot be an OLP polytope and is thus not subject to any of our general theorems. When n = 2 and n = 3, BD(n) is a reflexive polytope, but BD(3) does not have the IDP. Moreover, BD(4) fails to be reflexive.
Additionally, one could consider Birkhoff constructions for Coxeter groups of exceptional type, in particular E 6 , E 7 and E 8 (see, e.g., [BB05]). While we did not consider these polytopes in our investigation, we do raise the following question: Question 8.1. Do the Birkhoff polytope constructions for E 6 , E 7 , and E 8 have the IDP? Are these polytopes reflexive? Do they have other interesting properties? 8.2. Future directions. In addition to considering Birkhoff polytopes of other types and connections to Gale duality as discussed above, there are several immediate avenues for further research. Coxeter groups of great interest in the broader community of algebraic and geometric combinatorics (see, e.g., [BB05]). Subsequently, it is natural to consider how the Ehrhart-theoretic study of the type-B Birkhoff polytope informs research area. This leads to the following question: Question 8.2. Does the convex structure of BB(n) encode combinatorial or group theoretic structure of interest in Coxeter combinatorics?
An additional future direction is to consider applications of the orthant-lattice property, particularly those of Theorem 3.2 and Remark 3.5. One potentially fruitful avenue is an application to reflexive smooth polytopes. Recall that a lattice polytope P ⊂ R d is simple if every vertex of P is contained in exactly d edges (see, e.g., [Zie95]). A simple polytope P is called smooth if the primitive edge direction generate Z d at every vertex of P . Smooth polytopes are particularly of interest due to a conjecture commonly attributed to Oda [Oda]: Conjecture 8.3 (Oda). If P is a smooth polytope, then P has the IDP.
This conjecture is not only of interest in the context of Ehrhart theory, but also in toric geometry. One potential strategy is to consider similar constructions to OLP polytopes for smooth reflexive polytopes to make progress towards this problem. As a first step, we pose the following question: Question 8.4. Are all smooth reflexive polytopes OLP polytopes? Furthermore, regarding reflexive OLP polytopes one can ask the question: Question 8.5. Given a reflexive OLP polytope P , under what conditions can we guarantee that P * is a reflexive OLP polytope? By (2), this has a positive answer when P is an unconditional reflexive polytope. However, there are multiple examples of failure in general even in dimension 2 (see Figure 1).