Kneser’s Theorem and Inequalities in Ehrhart Theory

We demonstrate how additive number theory can be used to produce new classes of inequalities in Ehrhart theory. More specifically, we use a classical result of Kneser to produce new inequalities between the coefficients of the Ehrhart δ-vector of a lattice polytope. The inequalities are indexed by the vertices of rational polyhedra Q(r, s) ⊆ R for 0 ≤ r ≤ s. As an application, we deduce all possible ‘balanced’ inequalities between the coefficients of the Ehrhart δ-vector of a lattice polytope containing an interior lattice point, in dimension at most 6.