# Metric cotype

### 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.

## Authors

Manor Mendel

Computer Science Division
The Open University of Israel
43107 Raanana
Israel

Assaf Naor

Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States