Uniform normal structure and related notions

Let X be a Banach space, let φ denote the usual Kuratowski measure of noncompactness, and let kX (ε) = sup r (D) where r (D) is the Chebyshev radius of D and the supremum is taken over all closed convex subsets D of X for which diam (D) = 1 and φ (D) ≥ ε. The space X is said to have φ-uniform normal structure if kX (ε) < 1 for each ε ∈ (0, 1) . It is shown that this concept, which lies strictly between normal structure and uniform normal structure, implies reflexivity. Hence if X has then X has the fixed point property for nonexpansive mappings. Related concepts in metric spaces are also discussed.