Quantitative isoperimetry à la Levy-Gromov

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the
distance of isoperimetric sets from geodesic balls is quantitatively controlled
in terms of the gap between the isoperimetric profile of the manifold and that
of a round sphere of suitable radius. The deficit between the diameters of the
manifold and of the corresponding sphere is bounded likewise. These results are
actually obtained in the more general context of (possibly non-smooth) metric
measure spaces with curvature-dimension conditions through a quantitative
analysis of the transport-rays decompositions obtained by the localization
method.