Isoperimetry and Symmetrization for Sobolev spaces on metric spaces

TL;DR

This paper develops a unified framework using isoperimetry to derive symmetrization inequalities, enabling new insights into Sobolev inequalities and related principles in metric spaces.

Contribution

It introduces a novel approach connecting isoperimetry with symmetrization to study Sobolev inequalities in metric spaces.

Findings

New symmetrization inequalities derived from isoperimetry

Applications to concentration inequalities in metric spaces

Extensions of Pólya-Szegö and Faber-Krahn principles

Abstract

Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the P\'{o}% lya-Szeg\"{o} and Faber-Krahn principles.