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

Abstract

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 $M^{\ast }(K)$ 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 $M^{\ast }(K)\le C{n}^{3/4}{L}_K$. 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 $1\le p\le \sqrt{n}$ to an almost-sharp bound for an arbitrary $p\ge \sqrt{n}$. 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 $n$.

15 pages; added references, to appear in IMRN. See publisher's website for final version

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