Examples of IDP lattice polytopes with non-log-concave $h^*$-vector

TL;DR

This paper provides examples of IDP lattice polytopes with non-log-concave $h^*$-vectors, challenging the conjecture that such polytopes always have unimodal Ehrhart series coefficients.

Contribution

It presents the first known examples of IDP polytopes with non-log-concave $h^*$-vectors, addressing a long-standing open question.

Findings

IDP polytopes can have non-log-concave $h^*$-vectors

Answers a question posed by Ferroni and Higashitani

Provides initial examples for ongoing research

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 $h^*$-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that $h^*$-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.