On the mean-width of isotropic convex bodies and their associated L_p-centroid bodies

For any origin-symmetric convex body K in ℝⁿ in isotropic position, we obtain the bound M∗ (K) ≤ C √ nlog(n)²L_K, where M∗(K) denotes (half) the mean-width of K, L_K is the isotropic constant of K, and C >0 is a universal constant. This improves the previous best-known estimate M∗ (K) ≤Cn³/⁴L_K. Up to the power of the log(n) term and the L_K one, the improved bound is best possible, and implies that the isotropic position is (up to the L_K term) an almost 2-regular M-position. The bound extends to any arbitrary position, depending on a certain weighted average of the eigenvalues of the covariance matrix. Furthermore, the bound applies to the mean-width of L_p-centroid bodies, extending a sharp upper bound of Paouris for 1 ≤ p≤ √ n to an almost-sharp bound for an arbitrary p≥ √ n. The question of whether it is possible to remove the L_K term from the new bound is essentially equivalent to the Slicing Problem, to within logarithmic factors in n.