Thin-shell bounds via parallel coupling

TL;DR

This paper proves the thin-shell conjecture for high-dimensional log-concave distributions, showing most of the mass concentrates in a thin spherical shell, using novel coupling techniques and stochastic localization methods.

Contribution

It introduces a new coupling approach combining Eldan's stochastic localization and non-linear filtering to prove the thin-shell conjecture.

Findings

Most of the mass of log-concave vectors is in a thin shell

The shell width is proportional to 1/√n times the radius

Confirms the thin-shell conjecture in high dimensions

Abstract

We prove that for any log-concave random vector $X$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ \mathbb{E} (|X| - \sqrt{n})^2 \leq C $$ 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.