## Abstract

For any origin-symmetric convex body K in ℝ^{n} in isotropic position, we obtain the bound M∗ (K) ≤ C √ nlog(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) ≤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 ≤ 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.

Original language | English |
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Pages (from-to) | 3408-3423 |

Number of pages | 16 |

Journal | International Mathematics Research Notices |

Volume | 2015 |

Issue number | 11 |

DOIs | |

State | Published - 2015 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

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