A Monotonicity Property of h-vectors and h*-vectors

If Δ is a Cohen-Macaulay simplicial complex of dimension d - 1 and Δ′ is a Cohen-Macaulay subcomplex of dimension e - 1, such that no e + 1 vertices of Δ′ form a face of Δ, then we show that h (Δ′) ⩽ h(Δ), where h denotes the h-vector. In particular, h(Δ′) ⩽ h(Δ) if Δ and Δ′ are Cohen-Macaulay of the same dimension. Using similar techniques we obtain a class of Gorenstein complexes Δ, the h-vector of which is unimodal. Most of these results were obtained earlier by Kalai in a somewhat more complicated way. We then use our methods to give an analogous monotonicity property of Ehrhart polynomials of lattice polytopes (and more general objects). Our results on Ehrhart polynomials may be regarded as ʹlattice analoguesʹ of the well known monotonicity results concerning intrinsic volumes or quermassintegrals.