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, Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matousek.

46 pages. Fixes the layout

References in corpus (4)

• On metric Ramsey-type phenomena
• Euclidean quotients of finite metric spaces
• Finite metric spaces--combinatorics, geometry and algorithms
• Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

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• On the geometry of the countably branching diamond graphs
• A new coarsely rigid class of Banach spaces
• Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces $L_p$ and ${\ell }_p$ for $0<p<\infty$
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• Large scale geometry of metrisable groups
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• Snowflake universality of Wasserstein spaces
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• On the bi-Lipschitz geometry of lamplighter graphs
• A note on dichotomies for metric transforms
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• Improved bounds in the scaled Enflo type inequality for Banach spaces
• The Novikov Conjecture
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• Markov type and threshold embeddings
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