Radial continuous rotation invariant valuations on star bodies

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.

Two minor gaps and several typos corrected thanks to the referee

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