  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.
