The polytope algebra

Abstract

Let F be an ordered field, and let p denote the family of all convex polytopes in the d-dimensional vector space V over F . The universal abelian group ∏ corresponding to the translation invariant valuations on p has generators [P] for P ϵ p (with [⊘] = 0), satisfying the relations (V) [P ⌣ Q] + [P ⌢] = [P] + [Q] whenever P, Q, P ⌣ Q ϵ p , and (T) [ P + t] = [ P] for P ϵ p and t ϵ V. With multiplication induced by (M) [ P] · [ Q] = [ P + Q], ∏ is almost a graded commutative algebra over F , in that ∏ = ⊕ d r = 0 Ξ r , with Ξ 0 ≅ Z , Ξ r a vector space over F ( r ≥ 1), and Ξ r · Ξ s = Ξ r + s ( r, s ≥ 0, Ξ r = {0} for r > d). The dilatation (D) Δ( λ)[ P] = [ λP] for P ϵ p and λ ϵ F is such that Δ( λ) x = λ r x for x ϵ Ξ r and λ ≥ 0. Negative dilatations arise from the Euler map (E) [P] ↦ [P]∗ := ∑ F (−1) dimF [F] (the sum extending over all faces F of P), since Δ(λ)x = λ rx∗ for x ϵ Ξ r and λ < 0. Separating group homomorphisms for ∏ are the frame functionals, which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions. Two isomorphisms on ∏ are described: one related to cones of outer normal vectors, and the other to the polytope groups, obtained from ∏ by discarding polytopes of dimension less than d. Various applications of the polytope algebra are given, including a theory of mixed polytopes, which has implications for mixed valuations.