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. The reverse implication is in general false. It is shown that under a (possibly negative) lower bound condition on a natural notion of curvature associated to a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. The results are essentially the best possible (up to constants) and significantly extend all previously known results, which could deduce only dimension-dependent bounds or could not deduce anything stronger than a linear isoperimetric inequality in the restrictive nonnegative curvature setting. As a corollary, all of these previous results are recovered and extended by generalizing an isoperimetric inequality of Bobkov. Further applications will be described in subsequent works. 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.