Difference between families of weakly and strongly maximal integral lattice-free polytopes

TL;DR

This paper investigates the differences between two families of lattice-free polytopes in high dimensions, providing a super-exponential lower bound on the number of polytopes that are maximal in one family but not the other.

Contribution

It establishes a super-exponential lower bound on the count of polytopes that are maximal in the family of integral lattice-free polytopes but not in the larger family, highlighting structural differences.

Findings

For dimensions d ≥ 4, the family of maximal integral lattice-free polytopes is strictly smaller than the family of maximal lattice-free sets.

A super-exponential lower bound is derived for the number of polytopes in the difference set.

The result clarifies the structural gap between weakly and strongly maximal lattice-free polytopes in higher dimensions.

Abstract

A $d$-dimensional closed convex set $K$ in $\mathbb{R}^d$ is said to be lattice-free if the interior of $K$ is disjoint with $\mathbb{Z}^d$. We consider the following two families of lattice-free polytopes: the family $\mathcal{L}^d$ of integral lattice-free polytopes in $\mathbb{R}^d$ that are not properly contained in another integral lattice-free polytope and its subfamily $\mathcal{M}^d$ consisting of integral lattice-free polytopes in $\mathbb{R}^d$ which are not properly contained in another lattice-free set. It is known that $\mathcal{M}^d = \mathcal{L}^d$ holds for $d \le 3$ and, for each $d \ge 4$, $\mathcal{M}^d$ is a proper subfamily of $\mathcal{L}^d$. We derive a super-exponential lower bound on the number of polytopes in $\mathcal{L}^d \setminus \mathcal{M}^d$ (with standard identification of integral polytopes up to affine unimodular transformations).