DIAMOND GRAPHS AND SUPER-REFLEXIVITY

The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs D n Lipschitz embed into X with distortions independent of n. One of the consequences of that and previously known results is that dimension reduction a la Johnson–Lindenstrauss fails in any non-super-reflexive space with nontrivial type. We also introduce the concept of Lipschitz (p,r)-summing map and prove that every Lipschitz mapping is Lipschitz (p,r)-summing for every 1 ≤ r < p.