Radial continuous rotation invariant valuations on star bodies

TL;DR

This paper characterizes positive radial continuous rotation invariant valuations on star bodies in Euclidean space, showing they can be represented as integrals with respect to Lebesgue measure and approximated by dual quermassintegrals.

Contribution

It provides a complete integral representation of such valuations and demonstrates their approximation by linear combinations of dual quermassintegrals.

Findings

Valuations admit an integral representation with a continuous function.

Every valuation can be uniformly approximated by dual quermassintegrals.

The characterization applies to star bodies in rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m rf6m.

Abstract

We characterize the positive radial continuous and rotation invariant valuations $V$ defined on the star bodies of $\mathbb R^n$ as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is, $$V(K)=\int_{S^{n-1}}\theta(\rho_K)dm,$$ where $\theta$ is a positive continuous function, $\rho_K$ is the radial function associated to $K$ and $m$ is the Lebesgue measure on $S^{n-1}$. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.