Metric cotype

We introduce the notion of <i>metric cotype</i>, a property of metric spaces related to a property of normed spaces, called <i>Rademacher cotype</i>. 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 <i>F</i> is either almost universal (i.e., contains any finite metric space with any distortion > 1), or there exists α > 0, and arbitrarily large <i>n</i>-point metrics whose distortion when embedded in any member of <i>F</i> is at least Ω((log <i>n</i>)<sup>α</sup>). The same property is also used to prove strong non-embeddability theorems of <i>L</i><inf><i>q</i></inf> into <i>L</i><inf><i>p</i></inf>, when <i>q</i> > max{2, <i>p</i>}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus.

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