A quantitative result for the $k$-Hessian equation

TL;DR

This paper develops a symmetrization technique that refines inequalities related to the $k$-Hessian equation, providing quantitative improvements and applications to classical geometric inequalities.

Contribution

It introduces a symmetrization preserving mixed volume, leading to quantitative inequalities for the $k$-Hessian integral and related comparison results.

Findings

Quantitative improvement of Pólya-Szegő inequality for $k$-Hessian integrals

Quantitative inequality for solutions to the $k$-Hessian equation

Quantitative versions of Faber-Krahn and Saint-Venant inequalities

Abstract

In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a P\'olya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the P\'olya-Szeg\H o inequality for the $k$-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso \cite{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.