  Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove
that if a Riemannian manifold has Ricci curvature bounded below in a metric
ball which moreover has almost maximal volume, then in a smaller ball (in a
quantified sense) it holds an almost-euclidean isoperimetric inequality. The
result is actually established in the more general framework of non-smooth
spaces satisfying local Ricci curvature lower bounds in a synthetic sense via
optimal transportation.
