A proof of Lov\'asz's theorem on maximal lattice-free sets

TL;DR

This paper provides a new proof of Lovász's theorem on the structure of maximal lattice-free convex sets in Euclidean space, which are key in integer optimization, using Minkowski's fundamental theorem instead of Dirichlet's approximation.

Contribution

It offers an alternative proof of Lovász's theorem employing Minkowski's theorem, simplifying the understanding of maximal lattice-free sets in integer optimization.

Findings

Maximal lattice-free sets are polyhedra with at most 2^d facets.

Recession cones of these sets are spanned by integer vectors.

The proof enhances theoretical understanding in integer optimization.

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 cointain points of $\mathbb{Z}^d$ and $K$ is inclusion-maximal with respect to the above properties. A result of Lov\'asz assert that if $K$ is $d$-dimensional, then $K$ is a polyhedron with at most $2^d$ facets, and the recession cone of $K$ is spanned by vectors from $\mathbb{Z}^d$. A first complete proof of mentioned Lov\'asz's result has been published in a paper of Basu, Conforti, Cornu\'ejols 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 gemetry of numbers. We remark that the result of Lov\'asz is relevant in integer and mixed-integer optimization.