Continuity and representation of valuations on star bodies

Abstract

It is shown that every continuous valuation defined on the $n$-dimensional star bodies has an integral representation in terms of the radial function. Our argument is based on the non-trivial fact that continuous valuations are uniformly continuous on bounded sets. We also characterize the continuous valuations on the $n$-dimensional star bodies that arise as the restriction of a measure on ${\mathbb{R}}^n$.