Kneser's theorem and inequalities in Ehrhart theory

We demonstrate how additive number theory can be used to produce new classes of inequalities in the theory of enumeration of lattice points in polytopes. More specifically, we use a classical result of Kneser to produce new inequalities between the coefficients of the Ehrhart h^*-vector of a lattice polytope. As an application, we deduce all possible `balanced' inequalities between the coefficients of the Ehrhart h^*-vector of a lattice polytope containing an interior lattice point, in dimension at most 6.