A quantitative result for the k-Hessian equation

In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya–Szegő type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya–Szegő inequality for the k-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso (1989) 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.