Best possible lower bounds on the coefficients of Ehrhart polynomials

TL;DR

This paper establishes optimal lower bounds for the coefficients of Ehrhart polynomials of integral convex polytopes, extending known results to new coefficients and providing a comprehensive understanding of their minimal values.

Contribution

The authors prove that previously known bounds are also optimal for certain coefficients and introduce new bounds for others, advancing the theory of Ehrhart polynomial coefficients.

Findings

Bounds are optimal for coefficients r=3 and d-r even.

New best possible bounds are established for r=d-3.

Extends the range of coefficients with known optimal bounds.

Abstract

For an integral convex polytope $\mathcal{P} \subset \mathbb{R}^d$, we recall $L_\mathcal{P}(n)=|n\mathcal{P} \cap \mathbb{Z}^d|$ the Ehrhart polynomial of $\mathcal{P}$. Let $g_r(\mathcal{P})$ be the $r$th coefficients of $L_\mathcal{P}(n)$ for $r=0,\ldots,d$. Martin Henk and Makoto Tagami gave lower bounds on the coefficients $g_r(\mathcal{P})$ in terms of the volume of $\mathcal{P}$. They proved that these bounds are best possible for $r \in \{1,2,d-2\}$. We show that these bounds are also optimal for $r=3$ and $d-r$ even and we give a new best possible bound for $r=d-3$.