Inequalities for f *-vectors of lattice polytopes

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 -10,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.