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Convex sets with the Lipschitz fixed point property are compact. (English) Zbl 0566.47039
The authors have constructed a Lipschitz self mapping with fixed point. Using this result, they prove that a closed convex set in normed space has the fixed point property for Lipschitz maps if and only if it is compact. But the following problem is still open.
Problem. Let K be a weakly compact convex subset of \(c_ 0\). Does K have the fixed point property for uniformly Lipschitz maps?

47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
46A55 Convex sets in topological linear spaces; Choquet theory
Full Text: DOI
[1] Richard Arens, Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11 – 22. · Zbl 0046.11801
[2] Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), no. 3, 439 – 445. · Zbl 0518.46010
[3] F. Hausdorff, Erweiterung einer stetigen Abbildung, Fund. Math. 30 (1938), 40-47. · JFM 64.0621.03
[4] V. L. Klee Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30 – 45. · Zbl 0064.10505
[5] Jouni Luukkainen, Extension of spaces, maps, and metrics in Lipschitz topology, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 17 (1978), 62. · Zbl 0396.54025
[6] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837 – 842. · Zbl 0010.34606
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