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arXiv · 1404.7421

Lifting properties of maximal lattice-free polyhedra

Gennadiy Averkov·Amitabh Basu
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Abstract

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, Cornuéjols and Köppe~\cite{bcm} 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~\cite{bcm} 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.

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Gennadiy Averkov, Amitabh Basu. 2015-01-18. Lifting properties of maximal lattice-free polyhedra. https://arxiv.org/abs/1404.7421

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