Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

Abstract

Given an isotropic random vector X with log-concave density in Euclidean space \({\mathbb{R}^n}\) , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that $$\begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array}$$ 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 \({\sqrt{\mathbb{V}{\rm ar}(|X|)}}\) we obtain is of the order of n1/3, and improves down to n1/4 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.Lovasz-Simonovits is deduced.