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 is an element of [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 + root5)/2, improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces. (C) 2003 Elsevier Inc. All rights reserved.