# Maximal lattice-free convex sets in linear subspaces

## TL;DR

This paper explores the structure of maximal lattice-free convex sets within affine subspaces, demonstrating their polyhedral nature and extending classical results to broader contexts in integer programming.

## Contribution

It extends Lovász's theorem by characterizing maximal lattice-free convex sets as polyhedra in affine subspaces, relevant for integer programming inequalities.

## Key findings

- All irredundant inequalities derive from these sets.
- Maximal lattice-free convex sets are polyhedra.
- Extension of Lovász's theorem to affine subspaces.

## Abstract

We consider a model that arises in integer programming, and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lov\'asz characterizing maximal lattice-free convex sets in $\mathbb{R}^n$.