Uniform normal structure and related notions (Q2720308)

The authors introduce a notion which lies strictly between normal structure and uniform normal structure and implies reflexivity. Call \(\varphi\) the Kuratowski measure of noncompactness. A Banach space \(X\) is said to have \(\varphi\)-uniform normal structure if for each \(\varepsilon\in(0,1)\) we have that the supremum of the Chebyshev-centers of all closed bounded convex sets \(D\) with \(\text{diam}(D)=1\) and \(\varphi(D)\geq\varepsilon\) is smaller than 1. The main result of the article states that \(\varphi\)-uniform normal structure implies reflexivity and the fixed point property for nonexpansive mappings.