Modulus of convexity in Banach spaces

Let X be a Banach space, X-2 subset of or equal to X be a two-dimensional subspace of X, and S(X) = {x is an element of X, \\x\\ = 1} be the unit sphere of X. Let delta(epsilon) = inf{1 - \\x + y\\/2 : \\x - y\\ less than or equal to epsilon}, where x, y is an element of S (X-2) and 0 less than or equal to epsilon less than or equal to 2 is the modulus of convexity of X. The best results so far about the relationship between normal structure and the modulus of convexity of X are that for any Banach space X either delta(1) > 0 or delta(3/2) > 1/4 implies X has normal structure. We generalize the above results in this paper to prove that for any Banach space X, delta(1 + epsilon) > epsilon/2 for any epsilon, 0 less than or equal to epsilon less than or equal to 1, implies X has uniform normal structure. (C) 2003 Elsevier Science Ltd. All rights reserved.