Uniform normal structure and related notions.
W. A. Kirk, Brailey Sims · 2001
Let X be a Banach space, let $\phi$ denote the usual Kuratowski measure of noncompactness, and let $k_X (\epsilon) = sup r (D)$ 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 $\phi(D) \ge \epsilon$. The space $X$ is said to have $\phi$-uniform normal structure if $k_X (\epsilon) < 1$ for each $\epsilon \in (0, 1)$. It is shown that this concept, which lies strictly between normal structure and uniform normal structure, implies reflexivity. Hence if $X$ has $\phi$-uniform normal structure then $X$ has the fixed point property for nonexpansive mappings. Related concepts in metric spaces are also discussed.