The W*-convexity and normal structure in Banach spaces

Let X be a Banach space, S(X) = {x is an element of X : \\x\\ = 1} be the unit sphere of X. The parameter, modulus of W*-convexity, W*(epsilon) inf{<(x - y)/2, f(x)> : x,y is an element of S(X), \\x - y\\ greater than or equal to epsilon, f(x) is an element of del(x)}, where 0 less than or equal to epsilon less than or equal to 2 and del(x) subset of or equal to S(X*) be the set of norm 1 supporting functionals of S(X) at x, is investigated. The relationship among uniform nonsquareness, uniform normal structure and the parameter W*(c) are studied, and a known result is improved. The main result is that for a Banach space X, if there is epsilon, where 0 < c < 1/2, such that W-* (1 + epsilon) > epsilon/2 where W-* (1 + epsilon) lim(alpha-->epsilon)- W* (1 + alpha), then X has normal structure. (C) 2004 Elsevier Ltd. All rights reserved.