Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

We prove that if (Formula presented.) is a metric measure space with (Formula presented.) having (in a synthetic sense) Ricci curvature bounded from below by (Formula presented.) and dimension bounded above by (Formula presented.), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any (Formula presented.), (Formula presented.) and upper diameter bounds) holds, i.e. the isoperimetric profile function of (Formula presented.) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume (Formula presented.) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume (Formula presented.) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces. © 2016 Springer-Verlag Berlin Heidelberg