This is an announcement for the paper "Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures" by Olivier Guedon and Emanuel Milman.
Abstract: Given an isotropic random vector $X$ with log-concave density in Euclidean space $\Real^n$, we study the concentration properties of $|X|$. We show in particular that: [ \P(|X| \geq (1+t) \sqrt{n}) \leq \exp(-c n^{\frac{1}{2}} \min(t^3,t)) ;;; \forall t > 0 ~, ] for some universal constant $c>0$. This improves the best known deviation results above the expectation on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when $X$ is $\psi_\alpha$ ($\alpha \in (1,2]$), in precise agreement with the sharp Paouris estimates. The upper bound on the thin-shell width $\sqrt{\Var(|X|)}$ we obtain is of the order of $n^{1/3}$, and improves down to $n^{1/4}$ when $X$ is $\psi_2$. Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp Paouris large-deviation one.
Archive classification: math.FA
Remarks: 23 pages
Submitted from: emanuel.milman@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1011.1360
or