School of Mathematical Sciences
Monday, May 16, 2011
Schreiber 006, 12:15
(Technion, Israel Institute of Technology)
Isoperimetric and Concentration Inequalities - Equivalence and Applications
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