Related papers: A note on lattice-face polytopes and their Ehrhart…
We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov;…
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We…
A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…
Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new…
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…
We study semigroup algebras arising from lattice polytopes, compute their volume polynomials (particularizing work of Hochster), and establish strong Lefschetz properties (generalizing work of the first three authors). This resolves several…
Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which…
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its…
The local $h^*$-polynomial of a lattice polytope is an important invariant arising in Ehrhart theory. Our focus is on lattice simplices presented in Hermite normal form with a single non-trivial row. We prove that when the off-diagonal…
We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces,…
We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $\beta$-invariant of the matroid. This yields a lattice-point counting formula for the…
It is known that the Ehrhart polynomials of cross-polytopes, as well as of pyramids over them, have positive coefficients. We give a combinatorial proof of this fact by showing that a scaled version of the Ehrhart polynomials are generating…
We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n^2, where n is the…
We introduce a powerful connection between Ehrhart theory and additive number theory, and use it to produce infinitely many new classes of inequalities between the coefficients of the $h^*$-polynomial of a lattice polytope. This greatly…
As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side…
We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial…
For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube…
For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…
Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…