It was shown by Bobkov and the speaker that for a random vector X in $\mathbb{R}^n$ 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 that obtains an optimal concentration bound in this setting (optimal even in the constant terms, not just the exponent), using very simple techniques, and outline the proof. Applications that motivated the development of these results include high-dimensional convex geometry and random matrix theory— if time permits, we will outline these applications as well as extensions to the class of convex measures.Based on (multiple) joint works with Sergey Bobkov, Jiange Li, Matthieu Fradelizi, and Liyao Wang.
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