Geometric Analysis and Topology Seminar
On the Role of Convexity in Isoperimetry and Concentration, and a Sharp Quantitative Stability Result for the Spectral Gap of the Neumann Laplacian on Convex Domains
Speaker: Emanuel Milman, IAS
Location: Warren Weaver Hall 1013
Date: Friday, April 11, 2008, 2 p.m.
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 polynomial tail-decay of these functions, are all equivalent (to within universal constants, independent of the dimension). This 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 of the Cheeger constant for convex domains, as one over the expectation of the distance from the "worst" Borel set having half the volume of the domain. In addition, we easily recover (and extend) many previously known lower bounds, due to Payne-Weinberger, Li-Yau, Kannan-Lovász-Simonovits, Bobkov and Sodin, on the Cheeger constant of convex domains. 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-Émery.