### Abstract

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L_{p} coarsely or uniformly embeds into L_{q}. We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matoušek.

Original language | English (US) |
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Pages (from-to) | 247-298 |

Number of pages | 52 |

Journal | Annals of Mathematics |

Volume | 168 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2008 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Mathematics*,

*168*(1), 247-298. https://doi.org/10.4007/annals.2008.168.247