Super WCG Banach spaces

Abstract A Banach space X admits an equivalent strongly uniformly Gâteaux smooth norm if and only if it contains the dense range of a super weakly compact operator, which is equivalent to say that X is generated by a convex super weakly compact set. Moreover, if X is strongly generated by a convex super weakly compact set, then there is an equivalent norm on X such that its restriction to any reflexive subspace of X is both uniformly convex and uniformly Frechet smooth.