A Jordan-like decomposition theorem for valuations on star bodies

TL;DR

This paper proves a decomposition theorem for radial continuous valuations on star bodies, showing they can be expressed as differences of positive valuations, and characterizes rotationally invariant valuations via integral representations involving the radial function.

Contribution

It introduces a Jordan-like decomposition for valuations on star bodies and characterizes rotationally invariant valuations through integral formulas involving the radial function.

Findings

Decomposition of valuations into positive parts.

Characterization of rotationally invariant valuations.

Extension of previous positive valuation results.

Abstract

We show that every radial continuous valuation $V:\mathcal S_0^n\rightarrow \mathbb R$ 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 $\theta:[0,\infty)\rightarrow \mathbb R$ 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.