abstract: Formulas of Khovanskii-Pukhlikov, Brion-Vergne, and De Concini-Procesi-Vergne relate the volume with the number of integer points in a convex polytope. In this talk I will refine these formulas and talk about graded vector spaces that appear naturally in this context, the Dahmen-Micchelli spaces and their duals, the so-called P-spaces. It will turn out that the combinatorics of these spaces is determined by the underlying arithmetic matroid.