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

Abstract

Chaining techniques show that if X is an isotropic log-concave random vector in ℝn and Γ is a standard Gaussian vector then 𝔼||X|| ≤ Cn1/4𝔼||Γ|| 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 (formula presented). In particular, we show that if the thin-shell conjecture σn= O(1) holds, then n1/4 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.