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- W3099523897 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 $M^*(K)$ 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 $M^*(K) leq C n^{3/4} L_K$. 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 $1 leq p leq sqrt{n}$ 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 $n$." @default.
- W3099523897 created "2020-11-23" @default.
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- W3099523897 date "2014-03-18" @default.
- W3099523897 modified "2023-09-26" @default.
- W3099523897 title "On the Mean-Width of Isotropic Convex Bodies and their Associated Lp-Centroid Bodies" @default.
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- W3099523897 doi "https://doi.org/10.1093/imrn/rnu040" @default.
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