Polytopes, Valuations, and the Euler Relation

By a d-polytope we shall mean a d -dimensional convex polytope. We shall denote a j -dimensional face (or j-face ) of a polytope by F j . Every d -polytope P has proper j -faces for 0 ≦ j ≦ d — 1 and we shall also say that P is a d -face of itself. Observe that every face of a polytope is again a polytope. The collection of all convex polytopes shall be denoted by .