Mendel-Naor - cotype

Metric Cotype

[This gives the definitive answer to the question when Lp is uniformly (coarsely) embeddable into Lq.]

Title: Metric Cotype
Authors: Manor Mendel, Assaf Naor
Comments: 42 pages
Subj-class: Functional Analysis; Metric Geometry
MSC-class: 46B20; 51F99
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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 non-linear 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 non-linear analog of the
Maurey-Pisier theorem, and use it to answer a question posed by Arora,
Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain
quantitative bounds in a metric Ramsey theorem due to Matousek.