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 array \n-1{0, n-11, , n-1d\}, 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.