A quantitative result for the $k$-Hessian equation

In this paper, we consider a symmetrization with respect to mixed volumes of convex sets, for which a P\'olya-Szeg\"o type inequality holds. We improve the P\'olya-Szeg\"o for the $k$-Hessian integral in a quantitative way, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso for solutions to the $k$-Hessian equation. As an application of the first result we prove a quantitative version of the Faber-Krahn and Saint-Venant inequalities for these equations.