Normal structure and invariance of Chebyshev center under isometries (Q905987)

Let \(K\) be a nonempty, convex and weakly compact subset of a Banach space \(X\). \textit{M. S. Brodskij} and \textit{D. P. Mil'man} [Dokl. Akad. Nauk SSSR, n. Ser. 59, 837--840 (1948; Zbl 0030.39603)] proved that if \(K\) has normal structure, then there is a common fixed point for all surjective isometries of \(K\) which is located in the Chebyshev center \(C(K)\). In this paper it is shown that if the set of all directions in which \(X\) is not uniformly convex is contained in a countable union of its \(n\)-dimensional subspaces, then the same assertion holds for all isometries from \(K\) into \(K\), and that a normed space \(X\) with the above property has normal structure. This extends the results of \textit{T.-C. Lim} et al. [J. Math. Anal. Appl. 282, No. 1, 1--7 (2003; Zbl 1032.47036)] and \textit{M. A. Smith} [Math. Ann. 233, 155--161 (1978; Zbl 0391.46014)].