Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in ℝⁿ, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers ℝⁿ. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in ℝⁿ 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.