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).