The approximate fixed point property in Banach and hyperbolic spaces (Q1173833)

Let \(C\) be a closed convex subset of a Banach space \(X\). \(C\) is said to have the approximate fixed point property for nonexpansive mappings (AFPP) if for every nonexpansive mapping \(T: C\to C\), \(\inf\{| x- Tx|\mid\;x\subset C\}=0\). In the paper a geometrical characterization is given for those convex subsets of a Banach space (more generally a hyperbolic space) which possess the AFPP for nonexpansive mappings: Theorem. A convex subset \(C\subset X\) has the AFPP if and only if for every sequence \(\{x_ n\}\subset C\) such that \(| x_ n|\to\infty\) as \(n\to\infty\) and every \(f\in S(X^*)\) (the unit sphere of \(X^*\)) \(\lim_{n\to\infty}\sup f({x_ n \over | x_ n|})<1\).