A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

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.Comment: Published in the proceedings of 2017 Southern Regional Algebra Conferenc

The boundary volume of a lattice polytope

For a d-dimensional convex lattice polytope P, a formula for the boundary volume is derived in terms of the number of boundary lattice points on the first \floor{d/2} dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.Comment: 21 pages; subsumes arXiv:1002.1908 [math.CO]; to appear in the Bulletin of the Australian Mathematical Societ

A note on palindromic δ\delta-vectors for certain rational polytopes

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart $\delta$-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.Comment: 4 page

Convex and Algebraic Geometry

The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry

Convex Polytopes and Enumeration

AbstractThis is an expository paper on connections between enumerative combinatorics and convex polytopes. It aims to give an essentially self-contained overview of five specific instances when enumerative combinatorics and convex polytopes arise jointly in problems whose initial formulation lies in only one of these two subjects. These examples constitute only a sample of such instances occurring in the work of several authors. On the enumerative side, they involved ordered graphical sequences, combinatorial statistics on the symmetric and hyperoctahedral groups, lattice paths, Baxter, André, and simsun permutations,q-Catalan andq-Schröder numbers. From the subject of polytopes, the examples involve the Ehrhart polynomial, the permutohedron, the associahedron, polytopes arising as intersections of cubes and simplices with half-spaces, and thecd-index of a polytope