Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration

TL;DR

This paper investigates the stability and equivalence of various inequalities on Riemannian manifolds with measure, providing new bounds, perturbation results, and characterizations that deepen understanding of geometric and functional inequalities.

Contribution

It extends the Holley--Stroock perturbation lemma for log-Sobolev inequalities and introduces a new dimension-independent characterization of Transport-Entropy inequalities.

Findings

Stability of inequalities under measure perturbations with improved bounds

Equivalence of Transport-Entropy inequalities with different cost-functions

Dimension-independent characterization of Transport-Entropy inequalities without curvature assumptions

Abstract

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided $L^\infty$ bound on the ratio between their densities, Wasserstein distances, and Kullback--Leibler divergence. In particular, an extension of the Holley--Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.