Convex sets with the Lipschitz fixed point property are compact

Lin, P. K.; Sternfeld, Y.

doi:10.1090/S0002-9939-1985-0776193-5

Convex sets with the Lipschitz fixed point property are compact
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by P. K. Lin and Y. Sternfeld

Proc. Amer. Math. Soc. 93 (1985), 633-639

DOI: https://doi.org/10.1090/S0002-9939-1985-0776193-5

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Abstract:

Let $K$ be a noncompact convex subset of a normed space $X$. It is shown that if $K$ is not totally-bounded then there exists a Lipschitz self map $f$ of $K$ with $\operatorname {inf}\left \{ {\left \| {x - f\left ( x \right )} \right \|:x \in K} \right \} > 0$, while if $K$ is totally-bounded then such a map does not exist, but still $K$ lacks the fixed point property for Lipschitz mappings. It follows that a closed convex set in a normed space has the fixed point property for Lipschitz maps if and only if it is compact.

References

Richard Arens, Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11–22. MR 49543
Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), no. 3, 439–445. MR 699410, DOI 10.1090/S0002-9939-1983-0699410-7

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V. L. Klee Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30–45. MR 69388, DOI 10.1090/S0002-9947-1955-0069388-5
Jouni Luukkainen, Extension of spaces, maps, and metrics in Lipschitz topology, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 17 (1978), 62. MR 503040
E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. MR 1562984, DOI 10.1090/S0002-9904-1934-05978-0