Thin-shell bounds via parallel coupling
Boaz Klartag, Joseph Lehec
TL;DR
This work proves a universal thin-shell bound for isotropic, log-concave random vectors: $\mathrm{Var}(|X|^2) \le C n$, which implies that $|X|$ concentrates in a thin spherical shell of width $O(1/\sqrt{n})$ around $\sqrt{n}$. The authors develop a novel framework based on parallel coupling of log-affine perturbations (logarithmic tilts) of the law, leveraging Eldan's stochastic localization and non-linear filtering, together with Guan's growth-regularity technique. A key component is translating the $H^{-1}(\mu)$-norm of coordinate functions into a bound controlled by the covariance process $(A_t)$ associated with stochastic localization, and bounding the eigenvalues of $A_t$ via Guan-type estimates and stopping-time analysis. The results connect thin-shell concentration to the Gaussian approximation of marginals, provide a robust path to central limit theorems for convex bodies, and strengthen the link between thin-shell conjecture and Bourgain's slicing problem in high-dimensional convex geometry.
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.