Ehrhart-Macdonald reciprocity extended

Abstract

For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an intimate relation between these two counting functions. A similar counting function and reciprocity law exists for the sum of all solid angles at integer points in dilates of P. We derive a unifying generalization of these reciprocity theorems which follows in a natural way from Brion's Theorem on conic decompositions of polytopes.