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 RnRn as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is,V(K)=∫Sn−1θ(ρK)dm, where θ   is a positive continuous function, ρKρK is the radial function associated to K and m   is the Lebesgue measure on Sn−1Sn−1. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.