Unique Minimal Liftings for Simplicial Polytopes

TL;DR

This paper characterizes when minimal liftings of inequalities from maximal lattice-free simplicial polytopes are unique, linking geometric properties of simplices to the uniqueness of their minimal liftings in lattice-free convex analysis.

Contribution

It provides a characterization of the region where minimal liftings are unique and establishes a criterion for simplices with lattice vertices to have unique minimal liftings.

Findings

Region of unique minimal liftings characterized

Unique minimal lifting occurs iff all vertices are lattice points

Results apply to simplices with one lattice point per facet

Abstract

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.