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