2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69459In this note we point out the relation between Brion's formula for the lattice point generating function of a convex polytope in terms of the vertex cones [Brion1988] on the one hand, and the polar decomposition à la Lawrence/Varchenko [Lawrence1991, Varchenko1987] on the other. We then go on to prove a version of polar decomposition for non-simple polytopes.11 pages, 6 figures. Followed suggestions by referee and restructured section 3. Accepted for publicationCombinatorics52B45 (Primary); 11L03, 11P21 (Secondary)text