Bounding the norm of a log-concave vector via thin-shell estimates

TL;DR

This paper establishes a new inequality relating the norm of isotropic log-concave vectors to the thin-shell constant, offering insights into the slicing problem and convex geometry.

Contribution

It introduces a novel approach to bounding the norm of log-concave vectors using thin-shell estimates, connecting the thin-shell conjecture to geometric inequalities.

Findings

New inequality relating vector norms to thin-shell constant

Conditional result: thin-shell conjecture implies improved bounds

Provides alternative proof for the slicing problem

Abstract

Chaining techniques show that if X is an isotropic log-concave random vector in R^n and Gamma is a standard Gaussian vector then E |X| < C n^{1/4} E |Gamma| for any norm |*|, where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant sigma_n = sup ((var|X|^){1/2} ; X isotropic and log-concave on R^n). In particular, we show that if the thin-shell conjecture sigma_n = O(1) holds, then n^{1/4} can be replaced by log (n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a logarithmic factor.