## Abstract

For p ε (2;∞), the metric X_{p} inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this re-sult include the following sharp statements about embeddings of L_{q} into L_{p} when 2 < q < p <∞: the maximal θ ε (0, 1] for which L_{q} admits a bi-θ-Hölder embedding into L_{p} equals q/p, and for m, n ε N, the small-est possible bi-Lipschitz distortion of any embedding into L_{p} of the grid (1,...m)^{n}⊆ l^{n}_{q} is bounded above and below by constant multiples (de-pending only on p; q) of the quantity min(n^{(p-q)(q-2)/(q2(p-2)});m^{(q-2)/q}).

Original language | English (US) |
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Pages (from-to) | 991-1016 |

Number of pages | 26 |

Journal | Annals of Mathematics |

Volume | 184 |

Issue number | 3 |

DOIs | |

State | Published - 2016 |

## All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)

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