Thin-shell bounds via parallel coupling

Abstract

We prove that for any log-concave random vector $X$ in ${\mathbb{R}}^n$ with mean zero and identity covariance, $$\mathbb{E}(|X|-\sqrt{n}{)}^2\le C$$ where $C>0$ is a universal constant. Thus, most of the mass of the random vector $X$ is concentrated in a thin spherical shell, whose width is only $C/\sqrt{n}$ times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of $X$ related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture.

36 pages v2: a few typos fixed and some minor modifications in the introduction