Let P be a lattice polytope in R^n, and let P \cap Z^n = {v_1,...,v_N}. If the N + \binom N2 points 2v_1,...,2v_N; v_1+v_2,...v_{N-1}+v_N are distinct, we say that P is a "distinct pair-sum" or "dps" polytope. We show that, if P is a dsp polytope in R^n, then N \le 2^n, and, for every n, we construct dps polytopes in R^n which contain 2^n lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials.
The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.
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.
In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n…
We show how to compute the Ehrhart polynomial of the free sum of two lattice polytopes containing the origin $P$ and $Q$ in terms of the enumerative combinatorics of $P$ and $Q$. This generalizes work of Beck, Jayawant, McAllister, and Braun, and follows from the observation that the weighted $h^*$-polynomial is multiplicative with respect to the free sum. We deduce that given a lattice polytope $P$ containing the origin, the problem of computing the number of lattice points in all rational dilates of $P$ is equivalent to the problem of computing the number of lattice points in all integer dil…
We show that the Euler-MacLaurin formula for Riemann sums has an n-dimensional analogue in which intervals on the line get replaced by convex polytopes.
In this paper, we will describe the space spanned by the angle-sums of polytopes, recorded in the alpha-vector. We will consider the angles sums of simplices and the angles sums and face numbers of simplicial polytopes and general polytopes. We will construct families of polytopes whose angle sums span the spaces of polytopes defined by the Gram and Perles equations, analogues of the Euler and Dehn-Sommerville equations. We show that the dimensions of the affine span of the space of angle sums of simplices is floor[(d-1)/2] + 1, and that of the angle sums and face numbers of simplicial polytop…
Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope contains a finite set of lattice points corresponding to nonnegative integral linear combinations. The partition polytope associated to an integral $a$ is a rational convex polytope, and any rational convex polytope can be realized canonically as a pa…
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.
Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is triangulable, namely, every polytope can be decomposed into simplexes. Thus it may be possible to represent polytope numbers by sums of simplex numbers. We analyzes a special type of triangulation, called pointed triangulation, and develops several methods to represent polytope numbers by sums of simplex numbers.
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…
If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in $\mathbf{R}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the $h$-fold sumset $hP$ is not necessarily the sum of $h$ lattice points in $P$. It is proved that if the polytope $P$ is a union of unimodular simplices, then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
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…
We will study the angle sums of polytopes, listed in the $α$-vector, working to exploit the analogy between the f-vector of faces in each dimension and the alpha-vector of angle sums. The Gram and Perles relations on the $α$-vector are analogous to the Euler and Dehn-Sommerville relations on the f-vector. First we describe the spaces spanned by the the alpha-vector and the $α$-f-vectors of certain classes of polytopes. Families of polytopes are constructed whose angle sums span the spaces of polytopes defined by the Gram and Perles equations. This shows that the dimension of the affine span of…
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 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.
Let $\mathcal{P} \subset \mathbb{R}^{d}$ and $\mathcal{Q} \subset \mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\mathbb{R}^{d}$ and $\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\mathcal{P}$ and $\mathcal{Q}$ to possess the integer decomposition property will be presented.
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…
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…
The interior angle vector ($\widehatα$-vector) of a polytope is a metric analogue of the $f$-vector in which faces are weighted by their solid angle. For simplicial polytopes, Dehn-Sommerville-type relations on the $\widehatα$-vector were introduced by Sommerville (1927) and Höhn (1953). Camenga (2006) defined the $\widehatγ$-vector, a linear transformation analogous to the $h$-vector and conjectured it to be non-negative. Using tools from geometric and algebraic combinatorics, we prove this conjecture and show that the $\widehatγ$-vector increases in the first half and is flawless. In contras…
The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms…
For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $\mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumptio…
We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation parameters. One of the main results is an extension of Macdonald's solid angle quasipolynomial for rational polytopes to a real analytic function of the dilation parameter, for any real convex polytope.
Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant over the set ${\mathcal D}^{k,l}(m,n)$. This integer piecewise-linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matr…
We study the number of lattice points in integer dilates of the rational polytope $P = (x_1,...,x_n) \in \R_{\geq 0}^n : \sum_{k=1}^n x_k a_k \leq 1$, where $a_1,...,a_n$ are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers $a_1,...,a_n$, find the largest value of t (the Frobenius number) such that $m_1 a_1 + ... + m_n a_n = t$ has no solution in positive integers $m_1,...,m_n$. This is equivalent to the problem of finding the largest dilate tP such that the facet $\sum_{k=1}^n x_k a_k = t$ contains no …
Polytope numbers for a given polytope are an integer sequence defined by the combinatorics of the polytope. Recent work by H. K. Kim and J. Y. Lee has focused on writing polytope number sequences as sums of simplex number sequences. We focus on $r$-rectified simplices and show that the sequences for these polytopes can be written as alternating sums of simplex numbers analogous to the inclusion-exclusion given by the geometric process of $r$-rectification.
We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from cons…
Given a real closed polytope $P$, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula, to give a new algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of $P$. The combinatorics of the face poset of $P$ plays a central role in the description of the Fourier transform of $P$. We also obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers…
In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given.
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…
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 study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the res…
Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let $φ\colon V\rightarrow \CC $ be a homogeneous polynomial function of degree $d$ (i.e., an element of $\Sym^{d} (V^{*})$). For $q\in \ZZ_{>0}$ and any face $F$ of $P$, let $D_{φ,F} (q)$ be the sum of $φ$ over the lattice points in the dilate $qF$. We define a generating function $G_φ(q,y) \in \QQ [q] [y]$ packaging together the various $D_{φ,F} (q)$, and show that it satisfies a functional equation that simultaneously generalizes Ehrhart…
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.
The triangulations of point configurations which decompose as a free sum are classified in terms of the triangulations of the summands. The methods employ two new kinds of partially ordered sets to be associated with any triangulation of a point set with one point marked, the "web of stars" and the "stabbing poset". Triangulations of smooth Fano polytopes are discussed as a case study.
We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably c…
Given a finite set of lattice points, we compare its sumsets and lattice points in its dilated convex hulls. Both of these are known to grow as polynomials. Generally, the former are subsets of the latter. In this paper, we will see that sumsets occupy all the central lattice points in convex hulls, giving us a kind of approximation to lattice points in polytopes.
We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a convex lattice polytope. We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.
The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers. This problem also goes by the name of the `coin exchange problem'. Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topolo…
We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $χ$ is a coloring of the lattice points in the orthant such that the color $χ(a+b)$ depends only on the colors $χ(a)$ and $χ(b)$, then the number of colors of the lattice points in the dilation $nP$ of $P$ is for large $n$ given by a polynomial (or, for rational $P$, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanski\uı on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanski\uı's the…
An asymptotic expansion formula of Riemann sums over lattice polytopes is given. The formula is an asymptotic form of the local Euler-Maclaurin formula due to Berline-Vergne. The proof given here for Delzant lattice polytopes is independent of the local Euler-Maclaurin formula. But we use it for general lattice polytopes. As corollaries, an explicit formula for each term in the expansion over Delzant polytopes in two dimension and an explicit formula for the third term of the expansion for Delzant polytopes in arbitrary dimension are given. Moreover, some uniqueness results are given.
We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictl…
Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. It is has been used for constructing canonical models in the sense of Mori Theory [arXiv:2008.05814]. Based on the Fine interior, we propose here a modification of the original adjoint polytopes as defined in [arXiv:1105.2415], by defining the Fine adjoint polytope $P^{F(s)}$ of $P$ as consisting of the points in $P$ that have lattice distance at least $s$ to all valid inequalities for $P$. We obtain a Fine Polyhedral Adjunc…
Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices $Δ_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and $n$ are fixed, their flags and some of their face lattice structure. In particular, we derive a closed formula for their {\em exponential generating flag function}. These polytopes are simple, include both the simplex $Δ_{n-1}$ and the permutahedron $Π_{n-1}$, and form a Minkowski basis for more general permutahedra.
For $A\in\mathbb{Z}^{m\times n}$ we investigate the behaviour of the number of lattice points in $P_A(b)=\{x\in\mathbb{R}^n:Ax\leq b\}$, depending on the varying vector $b$. It is known that this number, restricted to a cone of constant combinatorial type of $P_A(b)$, is a quasi-polynomial function if b is an integral vector. We extend this result to rational vectors $b$ and show that the coefficients themselves are piecewise-defined polynomials. To this end, we use a theorem of McMullen on lattice points in Minkowski-sums of rational dilates of rational polytopes and take a closer look at the…
We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.
Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq k$. We consider the following conjecture of Alspach: For any cyclic group $\Z_n$ and any subset $A \subseteq \Z_n \setminus \{0\}$ with $s_k \neq 0$, it is possible to find an ordering of the elements of $A$ such that no two of its partial sums $s_i$ and $s_j$ are equal for $0 \leq i < j \leq k$. We show that Alspach's Conjecture holds for prime $n$ when $k \g…
Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in $\mathbb{R}^{d-1}$, they define a family of convex $(n-d)$-dimensional polytopes $\widehatΣ_{1}, \ldots, \widehatΣ_{n-d}$. The $1$-skeletons of this family of polytopes are the flip graphs of certain combinatorial configurations which generalize triangulations of $\text{conv} \mathcal{A}$. We restrict our attention to $d=2$. First, we relate the $1$-skeleton of the Minkow…
Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<α|A|$ and $α\ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in $\mathbb{R}^d$. As an application of the case $A=B$ we give a lower bound on $|A^+|+|A^\times|$, where $A^+$ is the set of sums of distinct elements of $A$ and $A^\times$ is …
