Examples of IDP lattice polytopes with non-log-concave $h^{\ast }$-vector

Abstract

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the $h^{\ast }$-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that $h^{\ast }$-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

4 pages, 1 figure