# Lifting properties of maximal lattice-free polyhedra

arXiv: https://arxiv.org/abs/1404.7421 · HTML: https://mlsanity.com/paper/1404.7421/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2014.

## Built on by (most cited first, 1 of 1)

- [Learning Cut Generating Functions for Integer Programming](https://mlsanity.com/paper/2405.13992/) (2024)

## Abstract (arXiv)

We study the uniqueness of minimal liftings of cut-generating functions obtained from maximal lattice-free polyhedra. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polyhedra. This generalizes a previous result by Basu, Cornu'ejols and K"oppe~citebcm for em simplicial maximal lattice-free polytopes, thus completely settling this fundamental question about lifting for maximal lattice-free polyhedra. We further give a very general iterative construction to get maximal lattice-free polyhedra with the unique-lifting property in arbitrary dimensions. This single construction not only obtains all previously known polyhedra with the unique-lifting property, but goes further and vastly expands the known list of such polyhedra. Finally, we extend characterizations from~citebcm about lifting with respect to maximal lattice-free simplices to more general polytopes. These nontrivial generalizations rely on a number of results from discrete geometry, including the Venkov-Alexandrov-McMullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces.