Unimodality questions for integrally closed lattice polytopes

TL;DR

This paper investigates the unimodality of Ehrhart delta-vectors for integrally closed lattice polytopes, proving it for lattice parallelepipeds and proposing a new triangulation approach for reflexive polytopes.

Contribution

It extends unimodality questions to all integrally closed lattice polytopes and establishes it for lattice parallelepipeds, introducing a novel triangulation method for reflexive polytopes.

Findings

Proved unimodality for lattice parallelepipeds.

Generalized unimodality question to all integrally closed lattice polytopes.

Suggested a new triangulation approach for reflexive polytopes.

Abstract

It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart delta-vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the delta-vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.