Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound

TL;DR

This paper demonstrates that under a curvature lower bound, general concentration inequalities imply isoperimetric inequalities on manifolds with density, establishing an equivalence that extends previous results and is based on geometric methods.

Contribution

It proves the equivalence between concentration and isoperimetric inequalities under curvature bounds, extending known results and providing a new geometric proof approach.

Findings

Dimension-independent isoperimetric bounds from weak concentration

Extension of Bobkov's isoperimetric inequality to broader settings

Geometric proof method based on Gromov and Morgan's approach

Abstract

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry--\'Emery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension \emph{independent} bounds. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifold-with-density setting by Morgan.