Inequalities for f *-vectors of lattice polytopes

Abstract

The Ehrhart polynomial ehr P (n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f *-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr P (n) with respect to the binomial coefficient basis $\begin{array}{} \bigl\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{d}\bigr\}, \end{array}$ where d = dim P. Similarly to h/h *-vectors, the f *-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f *-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f *-coefficients increases and the last quarter decreases. Even though f *-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h *-vector, there is a polytope with the same h *-vector whose f *-vector is unimodal.