Colloquium: Tuesday, April 25, 2 pm. Speaker: Emanuel Milman (Technion). Title: “The KLS isoperimetric conjecture – old and new”.
What is the optimal way to cut a convex bounded domain K in Euclidean space R^n into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovasz and Simonovits from the 90’s asserts that, if one does not mind gaining a constant numerical factor (independent of n) in the surface area, one might as well dissect K using a hyperplane. This conjectured essential equivalence between the former non-linear isoperimetric inequality and its latter linear relaxation has been shown over the last two decades to be of fundamental importance to the understanding of volumetric and spectral properties of convex domains.
Unfortunately, the KLS conjecture has only been established for a handful of families of convex bodies, such as unit-balls of \ell_p, convex bodies of revolution, Cartesian products thereof, and a few more families of log-concave measures. In this talk, we describe a recent joint work with Alexander Kolesnikov, in which we confirm the validity of the conjecture for the class of generalized Orlicz balls (satisfying a mild technical assumption), i.e. certain level sets of \sum_i V_i(x_i), where V_i are (one-dimensional) convex functions. A key feature of our approach is that no symmetry assumptions are required from V_i.
Our method is based on the equivalence between isoperimetry and concentration for log-concave measures, which reduces the KLS conjecture to a question about concentration of Lipschitz functions on K. We establish the latter concentration by successively transferring concentration (or large-deviation) information between several auxiliary measures we construct, using the various transference tools developed by the speaker over the past years.