In this paper we use a generating function approach to record and calculate entries of the Minkowski tensors of a polytope. We focus on ''surface tensors'', extending the methods used in arXiv:1807.10258 for moments of the uniform distribution which correspond to volume tensors. In this context we also extend the definition of the adjoint polynomial to the boundary complex of a polytope with simplicial facets. In the case of simplicial polytopes we give an explicit formulation for these surface tensors.
We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes P and Q, we show that P can be written as a Minkowski sum with a summand homothetic to Q if and only if P has the same number of vertices as it…
In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.
The Minkowski length of a lattice polytope $P$ is a natural generalization of the lattice diameter of $P$. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in $P$. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates $tP$ of a lattice polytope $P$ behaves polynomially in $t\in\mathbb{N}$. In this paper we prove that for any lattice polytope $P$, the Minkowski length of $tP$ for $t\in\mathbb{N}$ is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of c…
We provide a Minkowski sum decomposition of marked chain-order polytopes into building blocks associated to elementary markings and thus give an explicit minimal set of generators of an associated semi-group algebra. We proceed by characterizing the reflexive polytopes among marked chain-order polytopes as those with the underlying marked poset being ranked.
The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.
We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum o…
This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindstöm. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses th…
We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appear naturally in the representation theory of semi-simple Lie algebras, and hence we can give a necessary and sufficient condition on the marked poset such that the associated toric degenerations of the corresponding partial flag variety are isomorphic. We further show that the set of lattice points in such a marked poset polytope is the Minkowski sum of sets of lattice points for 0-1 polytopes. Moreov…
The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is satisfied by f-vectors of the sum and the summands. We study some of the implications of this equation.
Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point …
We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $Γ$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $Γ$. When $Γ$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $Γ$. These…
We consider an interesting class of combinatorial symmetries of polytopes which we call \emph{edge-length preserving combinatorial symmetries}. These symmetries not only preserve the combinatorial structure of a polytope but also map each edge of the polytope to an edge of the same length. We prove a simple sufficient condition for a polytope to realize all edge-length preserving combinatorial symmetries by isometries of ambient space. The proof of this condition uses Cauchy's rigidity theorem in an unusual way.
We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 a…
Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of $\mathfrak{gl}_n(\mathbb{C})$. The integer point transform of the Gelfand-Tsetlin polytope $\mathrm{GT}(λ)$ projects to the Schur function $s_λ$. Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials $\mathfrak{S}_{w}$ corresponding to Grassmannian permutations. For any permutation $w \in S_n$ with column-convex Rothe diagram, we construct a polytope $\mathcal{P}_{w}$ whose integer point transform project…
For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new ob…
This note provides a Lefschetz theorem for Minkowski sums of polytopes, and conclude lower bound theorems for Minkowski sums of polytopes. It is written as an appendix to arXiv:1405.7368, so notation and references follow that paper.
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The problems are grouped into the sections ``Coordinate Descriptions'', ``Combinatorial Structure'', ``Isomorphism'', ``Optimization'', ``Realizability'', and ``Beyond Polytopes''.
We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1\oplus{}P_2$, of two $d$-dimensional convex polytopes $P_1$ and $P_2$, as a function of the number of vertices of the polytopes. For even dimensions $d\ge{}2$, the maximum values are attained when $P_1$ and $P_2$ are cyclic $d$-polytopes with disjoint vertex sets. For odd dimensions $d\ge{}3$, the maximum values are attained when $P_1$ and $P_2$ are $\lfloor\frac{d}{2}\rfloor$-neighborly $d$-polytopes, whose vertex sets are chosen appropriately from two distinct $d$-dimensional moment-…
Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the Minkowski factorization of polytopes. Using our construction we show that for any given polytopal fan there is a polytope factorization basis, i.e. a finite set of polytopes with respect to which any polytope whose normal fan is refined by the original fan can be uniquely written as a signed Minkowski sum. We explicitly study the factorization of polymatroids a…
We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the polytopes, for any $d\ge 2$. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope $\mathcal{C}$, the problem of counting the number of $k$-faces of $P_1+P_2+P_3$, reduces to counting the number of $(k+2)$-faces of the subset of $\mathcal{C}$ comprising of the faces that contain at least one vertex from each $P_i$. In two dimensio…
We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the peri…
We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [2]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as $f$- and $h$-vector calculus and shellings, and generalizes the…
In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the upper bound~$O(n2^d)$ for the diameter of any normal pure simplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound $n-d$ if the complexes are, moreover, flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual …
In this paper we study $\times_0$-products of Lannér diagrams. We prove that every $\times_0$-product of at least four Lannér diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order $2$. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to $12$.
Given a lattice $L$, a full dimensional polytope $P$ is called a {\em Delaunay polytope} if the set of its vertices is $S\cap L$ with $S$ being an {\em empty sphere} of the lattice. Extending our previous work \cite{DD-hyp} on the {\em hypermetric cone} $HYP_7$, we classify the six-dimensional Delaunay polytopes according to their {\em combinatorial type}. The list of 6241 combinatorial types is obtained by a study of the set of faces of the polyhedral cone $HYP_7$.
To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced 'double posets', that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the 'double order polytope' and the 'double chain polytope' and we …
A simplicial polytope is combinatorially rigid if its combinatorial structure is determined by its graded Betti numbers which are important invariant coming from combinatorial commutative algebra. We find a necessary condition to be combinatorially rigid for 3-dimensional reducible simplicial polytopes and provide some rigid reducible simplicial polytopes.
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.
We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of poly…
Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytop…
Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck--Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by…
For a polytope we define the {\em flag polynomial}, a polynomial in commuting variables related to the well-known flag vector and describe how to express the the flag polynomial of the Minkowski sum of $k$ standard simplices in a direct and canonical way in terms of the {\em $k$-th master polytope} $P(k)$ where $k\in\nats$. The flag polynomial facilitates many direct computations. To demonstrate this we provide two examples; we first derive a formula for the $f$-polynomial and the maximum number of $d$-dimensional faces of the Minkowski sum of two simplices. We then compute the maximum discrep…
In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A double poset is a finite set equipped with two partial orders. To a double poset Chappell, Friedl and Sanyal (2017) associated the double order polytope. They determined the combinatorial structure for the class of compatible double posets. In this paper we generalize their description to all d…
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that t…
This thesis presents new applications of Gale duality to the study of polytopes, point configurations and oriented matroids with extremal combinatorial properties. The first part of the thesis explores construction techniques for neighborly polytopes and oriented matroids. First, we provide a new interpretation of Shemer's classical Sewing Construction for neighborly polytopes in terms of lexicographic extensions of oriented matroids. This allows us to provide a simplified proof and to generalize it to oriented matroids in two ways: the Extended Sewing Construction and the Gale Sewing Construc…
We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-t…
In this note we prove that the number of combinatorial types of $d$-polytopes with $d+1+α$ vertices and $d+1+β$ facets is bounded by a constant independent of $d$.
The combinatorial diameter of a polytope $P$ is the maximum value of a shortest path between two vertices of $P$, where the path uses the edges of $P$ only. In contrast to the combinatorial diameter, the circuit diameter of $P$ is defined as the maximum value of a shortest path between two vertices of $P$, where the path uses potential edge directions of $P$ i.e., all edge directions that can arise by translating some of the facets of $P$. In this paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the…
We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h*-vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley's nonnegativity and monotonicity of h*-vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our re…
Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag vector of the multiplex is computed, and shown to equal the flag vector of a many-folded pyramid over a polygon. Multiplexes, but not other ordinary polytopes, are shown to be elementary. It is shown that all complete subgraphs of the graph of a multiplex determine faces of the multiplex. The toric h-vectors of the ordinary 5-dimensional polytopes are given. G…
Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by $Φ_{k+r}(n_1,n_2,...,n_r)$, where $Φ_{\ell}(n_1,n_2,...,n_r)= \sum_{\substack{1\le{}s_i\le{}n_i s_1+...+s_r=\ell}} \prod_{i=1}^r\binom{n_i}{s_i}$, $\ell\ge{}r$. Fukuda and Weibel have also shown that the upper bound mentioned above is tight for $d\ge{}4$, $2\le{}r\le{}\lfloor\frac{d}{2}\rfloor$, and for a…
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover the polyto…
This paper was submitted to the Oberwolfach Conference "Combinatorial Convexity and Algebraic Geometry", October 1997. Let $M={\mathbb Z}^r$. For convex lattice polytopes $P,P'$ in ${\mathbb R}^r$, when is $(M \cap P)+ (M \cap P') = M \cap (P + P')$? Without any additional condition, the equality obviously does not hold. When the pair $(M,P)$ corresponds to a complex projective toric variety $X$ and an ample divisor $D$ on $X$, it is reasonable to assume that $P'$ corresponds to an ample (or, more generally, a nef) divisor $D'$ on the same $X$. Then the question correspons to the surjectivity …
We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne's generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes that were seemingly unapproachable. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan nu…
For a simplicial complex $Δ$, the graded Betti number $β_{i,j}(k[Δ])$ of the Stanley-Reisner ring $k[Δ]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $Δ$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $β_{k-1,k}(k[Δ])=(k-1)\binom{n-d}{k}$. We prove this combinatorially.
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that v(P)*f(P) is upper bounded by d*2^(d+1), for a large collection of families of such polytopes P. Here v(P) (resp. f(P)) is the number of vertices (resp. facets) of P, and d is its dimension. Whether this holds for all 2-level polytopes was asked in [Bohn et al., ESA 2015], a…
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts ab…
We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family $P$ is affinely reduced to a family $Q$ if for every polytope $p\in P$ there exists $q\in Q$ such that $p$ is affinely equivalent to $q$ or to a face of $q$, where $\dim q = O((\dim p)^k)$ for some constant $k$. Under this comparison the above-mentioned f…
For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric algebraic varieties.
