title: “The polytope algebra” - CiteKey: “McM89” - Type: journalArticle - Title: “The polytope algebra,” - Author: “McMullen, Peter;” - Year: 1989 - DOI: 10.1016/0001-8708(89)90029-7 - Collections: “Syllabus; Talbot 2022,”
The polytope algebra
Meta
- URL: https://www.sciencedirect.com/science/article/pii/0001870889900297
- URI: http://zotero.org/users/1049732/items/WL7PYTDJ
- Open in Zotero: Zotero
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 tϵV. With multiplication induced by (M) [P] · [Q] = [P + Q], ∏ is almost a graded commutative algebra over F, in that ∏ = ⊕dr = 0Ξr, with Ξ0 ≅ Z, Ξr a vector space over F (r ≥ 1), and Ξr · Ξs = Ξr + s (r, s ≥ 0, Ξr = {0} forr > d). The dilatation (D) Δ(λ)[P] = [λP] for P ϵ p and λ ϵ F is such that Δ(λ)x = λrx for xϵΞr 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 = λrx∗ for xϵΞr 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.