Spaces of small metric cotype

Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result.Comment: 14 pages, the needed isometric inequality now derived from the literature, rather than proved by hand; other minor typos and errors fixe

Metric Cotype

We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion > 1), or there exists α > 0, and arbitrarily large n-point metrics whose distortion when embedded in any member of F is at least Ω((log n)^α). The same property is also used to prove strong non-embeddability theorems of L_q into L_p, when q > max{2,p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus

Metric Cotype

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.Comment: 46 pages. Fixes the layou

Locally decodable codes and the failure of cotype for projective tensor products

It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product \ell_p\tp X fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then \ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a proved via a connection to the theory of locally decodable codes