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 $\mathbb{R}^{n}$ and Γ is a standard Gaussian vector then

$$\displaystyle{\mathbb{E}\Vert X\Vert \leq Cn^{1/4}\mathbb{E}\Vert \varGamma \Vert }$$

for any norm $\Vert \cdot \Vert$, where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant

$$\displaystyle{\sigma _{n} =\sup {\Bigl ( \sqrt{\mathrm{Var }(\vert X\vert )};\ X\text{ isotropic and log-concave on }\mathbb{R}^{n}\Bigr )}.}$$

In particular, we show that if the thin-shell conjecture σ _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.