This is an announcement for the paper "On the role of convexity in isoperimetry, spectral-gap and concentration" by Emanuel Milman.

Abstract: We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, Spectral-Gap of the Neumann Laplacian, Exponential concentration of 1-Lipschitz functions, and the a-priori weakest linear tail-decay of 1-Lipschitz functions, are all equivalent (to within universal constants). This substantially extends previous results of Maz'ya, Cheeger, Gromov--Milman, Buser and Ledoux. As an application, we conclude the stability of the Spectral-Gap for convex domains under convex perturbations which preserve volume (up to constants) and under maps which are ``on-average'' Lipschitz. We also easily recover (and extend) many previously known lower bounds, due to Payne--Weinberger, Li--Yau, Kannan--Lov'asz--Simonovits, Bobkov and Sodin, on the Cheeger constant for convex domains. We also provide a new characterization of the Cheeger constant, as one over the expectation of the distance from the ``worst'' Borel set having half the measure of the convex domain. As a by-product of our methods, we develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and q-capacities, the latter being notions introduced by Maz'ya and extended by Barthe--Cattiaux--Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. A crucial ingredient to our proof is a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\infty)$ curvature-dimension condition of Bakry-'Emery.

Archive classification: math.MG math.FA

Remarks: 70 pages, 1st version

The source file(s), Dingir120.eps: 7755 bytes

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