Tel-Aviv University
School of Mathematical Sciences

Department Colloquium

Monday, May 16, 2011

Schreiber 006, 12:15

Emanuel Milman

(Technion, Israel Institute of Technology)

Isoperimetric and Concentration Inequalities - Equivalence and Applications

Abstract: 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 that isoperimetric inequalities imply corresponding functional versions, which in turn imply concentration counterparts, but 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 independent bounds. Consequently, in such spaces, all of the above tiers of the hierarchy are equivalent.

Time permitting, we will mention several applications of this result, ranging from Statistical Mechanics to Spectral Geometry. We also derive new sharp isoperimetric inequalities, generalizing classical results due to P. Levy, Sudakov-Tsirelson and Borell, Gromov and Bakry-Ledoux, into one single form.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006