On a generalized James constant ✩

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a ∈ [0, 1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that J(X) < (1 + √5)/2, improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.  2003 Elsevier Inc. All rights reserved