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

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).

Proof of Lemma 11 fixed, Question 14 added