Smooth valuations on convex bodies and finite linear combinations of mixed volumes

Abstract

It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: every continuous, translation invariant, 𝑘-homogeneous valuation on convex bodies in ℝⁿ can be approximated uniformly on compact subsets by finite linear combinations of mixed volumes involving at most 𝑁ₙ,ₖ summands, where 𝑁ₙ,ₖ is a constant depending on 𝑛 and 𝑘 only. Moreover, 𝑛 − 𝑘 − 1 of the arguments of the mixed volumes can be chosen to be ellipsoids that do not depend on the valuation. The result is based on a corresponding description of smooth valuations in terms of finite linear combinations of mixed volumes.