Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound

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.