Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures

Details Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures

2010-11-03

arxiv.org/abs/1011.0943v3

Mathematics

Functional Analysis

29 pages - formulation is now general, estimating deviation of a linear image of X, and dependence on the \psi_\alpha constant

Scientific paper

Given an isotropic random vector $X$ with log-concave density in Euclidean space $\Real^n$, we study the concentration properties of $|X|$ on all scales, both above and below its expectation. We show in particular that: \[ \P(\abs{|X| -\sqrt{n}} \geq t \sqrt{n}) \leq C \exp(-c n^{1/2} \min(t^3,t)) \;\;\; \forall t \geq 0 ~, \] for some universal constants $c,C>0$. This improves the best known deviation results 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 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 large-deviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan--Lov\'asz--Simonovits is deduced.

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