Thin-Shell implies small-ball deviation via Gaussian tilts

Abstract

We show that uniform thin-shell estimates for isotropic log-concave measures $\mu$ on ${\mathbb{R}}^n$ yield precise and explicit deviation estimates for the Euclidean norm $|X|$ below the expectation, improving the square-root dependence of Klartag-Lehec to a quadratic one (which is best possible, up to numeric constants). Using the recent Chen-Klartag sharp variance bound, we deduce: $$\mu \left(|X|\le \sqrt{n}-s\right)\le \exp \left\{-\frac{s^2}{4}\left(1+O\left(\frac{s}{\sqrt{n}}\right)\right)\right\}\forall s\in (0,\sqrt{n}).$$ In particular, this yields a new and transparent proof that Thin-Shell implies Slicing (by passing through small-ball estimates). Our method is based on using central Gaussian tilts, recently introduced by Brazitikos, which may be thought of as a deterministic version of Eldan's stochastic localization. An anisotropic variant (when $\mu$ has general covariance structure) of these deviation estimates is also obtained.

18 pages, extended the results to the non-symmetric case and corrected various typos