On the $m$th-order Affine P\'olya-Szeg\"o Principle

TL;DR

This paper establishes a new affine Pólya-Szegö principle for affine energies, generalizing previous results and providing equality conditions, with multiple applications in analysis.

Contribution

It introduces a novel affine Pólya-Szegö principle for a family of energies, extending prior work and characterizing equality cases.

Findings

Recovers existing $L^p$ affine Pólya-Szegö principles as special cases.

Provides a new framework for affine energy inequalities.

Demonstrates various applications of the new principle.

Abstract

An affine P\'olya-Szeg\"o principle for a family of affine energies, with equality condition characterization, is demonstrated. In particular, this recovers, as special cases, the $L^p$ affine P\'olya-Szeg\"o principles due to Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao. Various applications of this new P\'olya-Szeg\"o principle are shown.