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• W2952943099 abstract "We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have emph{arbitrarily slow} uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, Gromov--Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are ``on-average'' Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the ``worst'' subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne--Weinberger, Li--Yau, Kannan--Lov'asz--Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry--Ledoux and 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." @default.
• W2952943099 created "2019-06-27" @default.
• W2952943099 creator A5052187675 @default.
• W2952943099 date "2007-12-26" @default.
• W2952943099 modified "2023-09-27" @default.
• W2952943099 title "On the role of Convexity in Isoperimetry, Spectral Gap and Concentration" @default.
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