Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three

A convex set with nonempty interior is maximal lattice-free if it is inclusion maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron P in ℝ^d is the smallest natural number s such that sP is an integral polyhedron. In this paper we show that, up to affine mappings preserving ℤ^d, the number of maximal lattice-free rational polyhedra of a given precision s is finite. Furthermore, we present the complete list of all maximal lattice-free integral polyhedra in dimension three. Our results are motivated by recent research on cutting plane theory in mixed-integer linear optimization.