The Steiner rearrangement in any codimension

TL;DR

This paper investigates the Steiner rearrangement in arbitrary codimension for Sobolev and BV functions, establishing inequalities and conditions for equality cases, thus advancing geometric analysis techniques.

Contribution

It introduces a Pólya-Szegő inequality for convex integrals and minimal conditions for equality cases in Steiner rearrangement in any codimension.

Findings

Proved a Pólya-Szegő inequality for convex integrals

Identified minimal assumptions for equality cases

Extended Steiner rearrangement analysis to arbitrary codimension

Abstract

We analyze the Steiner rearrangement in any codimension of Sobolev and $BV$ functions. In particular, we prove a P\'olya-Szeg\H{o} inequality for a large class of convex integrals. Then, we give minimal assumptions under which functions attaining equality are necessarily Steiner symmetric.