  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.
