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Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integerpoints) and their applications for cutting-plane methods in mixed-integeroptimization have been studied in recent literature. Notably, the family of allintegral lattice-free polyhedra which are not properly contained in anotherintegral lattice-free polyhedron has been of particular interest. We call thesepolyhedra $\mathbb{Z}^d$-maximal.  It is known that, for fixed $d$, the family $\mathbb{Z}^d$-maximal integrallattice-free polyhedra is finite up to unimodular equivalence. In view ofpossible applications in cutting-plane theory, one would like to have aclassification of this family. However, this turns out to be a challenging taskalready for small dimensions.  In contrast, the subfamily of all integral lattice-free polyhedra which arenot properly contained in any other lattice-free set, which we call$\mathbb{R}^d$-maximal lattice-free polyhedra, allow a rather simple geometriccharacterization. Hence, the question was raised for which dimensions thenotions of $\mathbb{Z}^d$-maximality and $\mathbb{R}^d$-maximality areequivalent. This was known to be the case for dimensions one and two. On theother hand, Nill and Ziegler (2011) showed that for dimension $d \ge 4$, thereexist 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 $\mathbb{R}^3$-maximalityand $\mathbb{Z}^3$-maximality are equivalent. As a consequence, theclassification of all $\mathbb{R}^3$-maximal integral polyhedra by Averkov,Wagner and Weismantel (2011) contains all $\mathbb{Z}^3$-maximal integralpolyhedra.

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