Continuity and representation of valuations on star bodies

TL;DR

This paper proves that continuous valuations on n-dimensional star bodies can be represented as integrals involving the radial function and characterizes those that are restrictions of measures on ^n.

Contribution

It establishes an integral representation for continuous valuations on star bodies and characterizes which arise from measures, advancing the understanding of valuation theory in convex geometry.

Findings

Continuous valuations are uniformly continuous on bounded sets.

Every continuous valuation has an integral representation via the radial function.

Characterization of valuations that are restrictions of measures on ^n.

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 $\mathbb R^n$.