Mokshay Madiman, University of Delaware

Abstract: It was shown by Bobkov and the speaker that for a random vector X in R^n drawn from a log-concave density e^{-V}, the information content per coordinate, namely V(X)/n, is highly concentrated about its mean. Their argument was nontrivial, involving the localization technique, and also gave suboptimal exponents, but it was sufficient to demonstrate that high-dimensional log-concave measures are in a sense close to uniform distributions on the annulus between 2 nested convex sets (generalizing the well known fact that the standard Gaussian measure is concentrated on a thin spherical annulus). We will present recent work with M. Fradelizi and Liyao Wang that obtains an optimal concentration bound in this setting (optimal even in the constant terms, not just the exponent), using much simpler techniques, and outline the proof. If time permits, we will discuss applications to high-dimensional convex geometry, and an extension to the larger class of convex measures obtained with M. Fradelizi and Jiange Li.