### 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) |
---|---|

Pages (from-to) | 247-298 |

Number of pages | 52 |

Journal | Annals of Mathematics |

Volume | 168 |

Issue number | 1 |

State | Published - Jul 2008 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Metric cotype.** / Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Annals of Mathematics*, vol. 168, no. 1, pp. 247-298.

}

TY - JOUR

T1 - Metric cotype

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2008/7

Y1 - 2008/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=49749093965&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49749093965&partnerID=8YFLogxK

M3 - Article

VL - 168

SP - 247

EP - 298

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -