From Area Measures to Valuations

Abstract

In Chap. 3, we studied the mixed volume as a real-valued functional on n-tuples of convex bodies in ${\mathbb R}^n$. Already the recursive definition of the mixed volume of polytopes P ₁, …, P _n involves the support function of one of the bodies, say P _n, and mixed functionals of the facets with the same exterior unit normal vector of the remaining bodies P ₁, …, P _n−1. This defining relation will lead to a fundamental and general relation between the mixed volume of convex bodies K ₁, …, K _n, the support function of one of the bodies, say K _n, and (what we shall call) the mixed area measure of the remaining convex bodies K ₁, …, K _n−1. Specializing these mixed area measures, we shall obtain the area measures S _j(K, ⋅), j ∈{0, …, n − 1}, of a single convex body K.