Thin-Shell implies small-ball deviation via Gaussian tilts

Abstract: We show that uniform thin-shell estimates for (even) isotropic log-concave measures $μ$ on $\mathbb{R}<sup>n$ 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: $$ μ\left(|X|\le \sqrt{n}-s\right) \le \exp\left{ -\frac{s^2}{4} \left(1+O\left(\frac{s}{\sqrt{n}}\right)\right) \right} \qquad \forall\, s\in(0,\sqrt{n}). $$ 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.