Reflexive Banach spaces not isomorphic to uniformly convex spaces

Clarkson 3 introduced the notion of uniform convexity of a Banach space: B is uniformly convex if for each e with 0<e^2 there is a 6(e) >0 such that whenever ||&|| = ||&'|| =1 and ||&-&'||^, then ||j+6 / ||^2(l-6()). Milman 4 and Pettis 5 have demonstrated that any uniformly convex space is reflexive ; 6 that is, that for each ji?** there is a &U with /3 ()=(&) for every 5*. The same result clearly holds if B is not uniformly convex but can be given a new norm defining the same topology under which the space is uniformly convex. It has been conjectured that every reflexive space can be given such a topologically equivalent uniformly convex norm ; that is, that, in Banach's terminology, 7 every reflexive space is isomorphic to a uniformly convex space. We shall show by a large class of examples that this is not the case; in fact the following result holds: THEOREM 1. There exist Banach spaces which are separable, reflexive, and strictly convex? but are not isomorphic to any uniformly convex space.