  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.

Functional Analysis;Metric Geometry;Probability