Symmetrization with respect to mixed volumes

TL;DR

This paper introduces a new symmetrization method based on mixed volume and anisotropic curvature, extending previous work, and demonstrates its effectiveness in inequalities and PDE comparison principles.

Contribution

It develops a novel symmetrization with respect to mixed volume, generalizing existing methods, and applies it to anisotropic Hessian equations and Sobolev inequalities.

Findings

Symmetrization diminishes anisotropic Hessian integrals.

Establishes a comparison principle for anisotropic Hessian equations.

Proves sharp anisotropic Sobolev inequalities.

Abstract

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