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uniformly convex space; bead space; central point
The notion of a metric bead space was introduced in the preceding paper (L. Pasicki: Bead spaces and fixed point theorems, Topology Appl., vol. 156 (2009), 1811–1816) and it was proved there that every bounded set in such a space (provided the space is complete) has a unique central point. The bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces. It appears that normed bead spaces are identical with uniformly convex spaces. On the other hand the "metric" approach leads to new elementary conditions equivalent to the uniform convexity. The initial part of the paper contains the proof that discus spaces (they seem to have a richer structure) are identical with bead spaces.
[1] Lim, T. C.: On asymptotic centers and fixed points of nonexpansive mappings. Canad. J. Math. 32 (1980), 421-430. DOI 10.4153/CJM-1980-033-5 | MR 0571935 | Zbl 0454.47045
[2] Pasicki, L.: A basic fixed point theorem. Bull. Polish Acad. Sci. Math. 54 (2006), 85-88. DOI 10.4064/ba54-1-8 | MR 2270797 | Zbl 1105.54022
[3] Pasicki, L.: Bead spaces and fixed point theorems. Topology Appl. 156 (2009), 1811-1816. DOI 10.1016/j.topol.2009.03.042 | MR 2519217 | Zbl 1171.54024
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