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It is well known that isoperimetric inequalities imply in a very generalmeasure-metric-space setting appropriate concentration inequalities. The formerbound the boundary measure of sets as a function of their measure, whereas thelatter bound the measure of sets separated from sets having half the totalmeasure, as a function of their mutual distance. We show that under a lowerbound condition on the Bakry--\'Emery curvature tensor of a Riemannian manifoldequipped with a density, completely general concentration inequalities implyback 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 ofBobkov, and provide a new proof that under natural convexity assumptions,arbitrarily weak concentration implies a dimension independent linearisoperimetric inequality. Further applications will be described in asubsequent work. Contrary to previous attempts in this direction, our method isentirely geometric, continuing the approach set forth by Gromov and adapted tothe manifold-with-density setting by Morgan.

Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound