Notions of maximality for integral lattice-free polyhedra: the case of dimension three

Lattice-free sets (convex subsets of ${\mathbb{R}}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra ${\mathbb{Z}}^d$-maximal. It is known that, for fixed $d$, the family ${\mathbb{Z}}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutting-plane theory, one would like to have a classification of this family. However, this turns out to be a challenging task already for small dimensions. In contrast, the subfamily of all integral lattice-free polyhedra which are not properly contained in any other lattice-free set, which we call ${\mathbb{R}}^d$-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of ${\mathbb{Z}}^d$-maximality and ${\mathbb{R}}^d$-maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, Nill and Ziegler (2011) showed that for dimension $d\ge 4$, there exist polyhedra which are ${\mathbb{Z}}^d$-maximal but not ${\mathbb{R}}^d$-maximal. In this article, we consider the remaining case $d=3$ and prove that for integral polyhedra the notions of ${\mathbb{R}}^3$-maximality and ${\mathbb{Z}}^3$-maximality are equivalent. As a consequence, the classification of all ${\mathbb{R}}^3$-maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all ${\mathbb{Z}}^3$-maximal integral polyhedra.