Poincar{\'e} and logarithmic sobolev inequalities for nearly radial measures

TL;DR

This paper investigates Poincaré and logarithmic Sobolev inequalities for nearly radial measures, extending known results and providing sharper bounds through various methods, with applications to log-concave measures.

Contribution

It introduces new bounds for logarithmic Sobolev inequalities in the radial case, using multiple approaches, and refines existing bounds for log-concave measures.

Findings

Derived sharp bounds for radial measures

Extended Poincaré inequality results to logarithmic Sobolev inequalities

Refined bounds for log-concave measures by Lee-Vempala

Abstract

If Poincar{\'e} inequality has been studied by Bobkov for radial measures, few is known about the logarithmic Sobolev inequalty in the radial case. We try to fill this gap here using different methods: Bobkov's argument and super-Poincar{\'e} inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee-Vempala in the logconcave bounded case are refined for radial measures.