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Isoperimetric and concentration inequalities - equivalence and applications

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Given a metric space equipped with a measure, various ways exist for studying the interaction between measure and metric. A very strong form is given by isoperimetric inequalities, which for a set of given measure, provide a lower bound on its boundary measure. A much weaker form is given by concentration inequalities, which quantify large-deviation behavior of measures of separated sets.

There are also other tiers, interpolating between these two extremes, such as the tier of Sobolev-type inequalities. It is classical (by results of Cheeger, Maz’ya, Gromov—Milman andothers) that isoperimetric inequalities imply corresponding functional versions, which in turn imply concentration counterparts.

However, in general, these implications cannot be reversed. We show that under a suitable (possibly negative) lower bound on the generalized Ricci curvature of a Riemannian-manifold-with-density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension mph{independent} bounds. Consequently, in such spaces, all of the above tiers of the hierarchy are equivalent, clarifying the role which curvature plays in the interaction between measure and metric. Several applications of this equivalence will be presented, ranging from Statistical Mechanics to Spectral Geometry. Time permitting, we will also present new mph{sharp} isoperimetric inequalities, generalizing classical results due to P. L’evy, Sudakov—Tsirelson and Borell, Gromov and Bakry—Ledoux, into one single form.

The talk will be self-contained and accessible to all.

This talk is part of the Isaac Newton Institute Seminar Series series.

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