On a generalized James constant

Abstract

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.