A Jordan-like decomposition theorem for valuations on star bodies

Abstract

We show that every radial continuous valuation $V:{\mathcal{S}}_0^n\to \mathbb{R}$ defined on the $n$-dimensional star bodies ${\mathcal{S}}_0^n$, and verifying $V(\{0\})=0$, can be decomposed as a sum $V=V^{+}-V^{-}$, where both $V^{+}$ and $V^{-}$ are positive radial continuous valuations on ${\mathcal{S}}_0^n$ with $V^{+}(\{0\})=V^{-}(\{0\})=0$. As an application, we show that radial continuous rotationally invariant valuations $V$ on ${\mathcal{S}}_0^n$ can be characterized as the applications on star bodies which can be written as $$V(K)={\int }_{S^{n-1}}\theta ({\rho }_K)dm,$$ where $\theta :[0,\infty )\to \mathbb{R}$ is a continuous function, ${\rho }_K$ is the radial function associated to $K$ and $m$ is the Lebesgue measure on $S^{n-1}$. This completes recent work of the second named author, where an analogous result is proved for the case of {\em positive} radial continuous rotationally invariant valuations.

The results in this paper have been subsumed in arXiv:1611.03345