  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.
