A proof of Lovász’s theorem on maximal lattice-free sets

Abstract

Let K be a maximal lattice-free set in ${\mathbb{R}^d}$ , that is, K is convex and closed subset of ${\mathbb{R}^d}$ , the interior of K does not contain points of ${\mathbb{Z}^d}$ and K is inclusion-maximal with respect to the above properties. A result of Lovász asserts that if K is d-dimensional, then K is a polyhedron with at most 2^d facets, and the recession cone of K is a linear space spanned by vectors from ${\mathbb{Z}^d}$ . A first complete proof of mentioned Lovász’s result has been published in a paper of Basu, Conforti, Cornuéjols and Zambelli (where the authors use Dirichlet’s approximation as a tool). The aim of this note is to give another proof of this result. Our proof relies on Minkowki’s first fundamental theorem from the geometry of numbers. We remark that the result of Lovász is relevant in integer and mixed-integer optimization.