We study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice-free polytopes. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polytopes. This generalizes a previous result by Basu, Cornuejols and Koeppe for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question about lifting. We also extend results from for minimal liftings in maximal lattice-free simplices to more general polytopes. These nontrivial generalizations require the use of deep theorems from discrete geometry and geometry of numbers, such as the Venkov-Alexandrov-McMullen theorem on translative tilings, and McMullen's characterization of zonotopes.
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