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Chaining techniques show that if X is an isotropic log-concave random vectorin 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 differentargument we establish a similar inequality relying on the thin-shell constantsigma_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 dualmean-width and the isotropic constant of an isotropic convex body.  In particular, we give an alternative proof of the fact that a positiveanswer to the thin-shell conjecture implies a positive answer to the slicingproblem, up to a logarithmic factor.

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