Symmetrization with respect to mixed volumes

In this paper we introduce a new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one quermassintegral due Talenti [26] and Tso [28]. We show Pólya-Szegő type principle for such — it diminishes Hessian integral quasi-convex functions. achieve by systematic study of invariants on non-symmetric matrices real eigenvalues higher order mean curvatures level sets, may be independent interest. As applications, establish comparison equations sharp Sobolev inequalities.