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 $\mathbb{R}^n$ can be approximated uniformly on compact subsets by finite linear combinations of mixed volumes involving at most $N_{n,k}$ summands, where $N_{n,k}$ 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.