h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a unimodular triangulation satisfies McMullen’s g-theorem; in particular it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. In the case of a polytopal Gorenstein normal monoid E(P), this technique reduces all questions about the Ehrhart h-vector to a normal Gorenstein polytope Q with exactly one interior lattice point. (These are the normal ones among the reflexive polytopes considered in connection with mirror symmetry.) If P has a unimodular triangulation, then it follows readily that the Ehrhart hvector of P coincides with the h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.