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For any origin–symmetric convex body  K  in   $\mathbb {R}^n$   in isotropic position, we obtain the bound   $M^*(K) \leq C \sqrt {n} \log (n)^2 L_K$  , where  M *( K ) denotes (half) the mean-width of  K ,  L   K   is the isotropic constant of  K , and  C &gt;0 is a universal constant. This improves the previous best-known estimate  M *( K )≤ Cn  3/4  L   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   $1 \leq p \leq \sqrt {n}$   to an almost-sharp bound for an arbitrary   $p \geq \sqrt {n}$  . 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 .

On the Mean-Width of Isotropic Convex Bodies and their Associated L p -Centroid Bodies