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Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we provethat if a Riemannian manifold has Ricci curvature bounded below in a metricball which moreover has almost maximal volume, then in a smaller ball (in aquantified sense) it holds an almost-euclidean isoperimetric inequality. Theresult is actually established in the more general framework of non-smoothspaces satisfying local Ricci curvature lower bounds in a synthetic sense viaoptimal transportation.

Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds