Radial continuous rotation invariant valuations on star bodies

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, $V(K)=\int_{S^{n-1}}θ(ρ_K)dm,$ where $θ$ is a positive continuous function, $ρ_K$ 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.

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