Normal Structure and Modulus of U -convexity in Banach Spaces

Let X be a normed linear space, and let S(X) = fx 2 X : kxk = 1g be the unit sphere of X. Brodskii and Milman 1] introduced the following geometric concept: Deenition 1. A bounded, convex subset K of a Banach space X is said to have normal structure if every convex subset H of K that contains more than one point contains a point x 0 2 H, such that supfkx 0 ? yk : y 2 Hg < d(H); where d(H) = supfkx?yk: x; y 2 Hg denotes the diameter of H. A Banach space X is said to have normal structure if every bounded, convex subset of X has normal structure. A Banach space X is said to have w-normal structure if every weakly compact set K in X that contains more than one point has normal structure. It is clear that if X is reeexive, then the normal structure and the w-normal structure coincide. Kirk 6] proved that if a weakly compact subset K of X has normal structure, then every nonexpansive mapping on K has a xed point. It is well known that uniformly convex Banach spaces 2], uniformly smooth spaces 8], the Banach spaces which are uniformly convex in every direction 9], the nearly uniformly convex spaces 3], U-spaces 4], and the Banach spaces with J (X) < 3 2 , where J (X) = supfky ? xk ^ ky + xk : x; y 2 S(X)g 4], have normal structure. Gao and Lau 4] showed that if X fails to have w-normal structure, then X satises geometric conditions given in the following lemma: 195