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.