2005/06/30 by Manor Mendel, Assaf Naor · 2 citations
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Banach space #Computer science #Discrete mathematics #Mathematics #Metric (unit) #Metric space #Nonlinear system #Pure mathematics #Space (punctuation) #math.FA #math.MG #msc:46B20 #msc:51F99
paper · pdf · doi:10.4007/annals.2008.168.247
published as Ann. of Math. 168(1), 247-298 (2008) · 46 pages. Fixes the layout
openalex publication_date 2008/07/01 · arxiv created 2011/02/08 · arxiv updated 2012/11/15 · openalex created_date 2020/07/02 · openalex updated_date 2026/08/05
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, Lovsz, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matouek.