Isoperimetry and symmetrization for logarithmic Sobolev inequalities

TL;DR

This paper introduces a unified approach using isoperimetry and symmetrization to analyze classical and logarithmic Sobolev inequalities, leading to new Gaussian symmetrization results and broad applicability.

Contribution

It develops a general framework connecting isoperimetry, symmetrization, and Sobolev inequalities, including novel Gaussian symmetrization inequalities.

Findings

New Gaussian symmetrization inequalities derived

Unified framework for classical and logarithmic Sobolev inequalities

Methods adaptable to various mathematical contexts

Abstract

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.