On counterexamples to a conjecture of Wills and Ehrhart polynomials whose roots have equal real parts

TL;DR

This paper investigates counterexamples to Wills' conjecture on Ehrhart coefficients of symmetric lattice polytopes, identifying specific families that satisfy or violate the conjecture, and explores related polytopes with roots sharing real parts.

Contribution

The paper provides counterexamples to Wills' conjecture and introduces a family of lattice polytopes related to l-reflexive polytopes that meet the conjectured inequalities.

Findings

Counterexamples to Wills' conjecture are presented.

A family of polytopes satisfying the conjecture is identified.

Connections to l-reflexive polytopes are established.

Abstract

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 length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.