Convex sets with the Lipschitz fixed point property are compact

Authors:
P. K. Lin and Y. Sternfeld

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

MSC:
Primary 47H10; Secondary 46B20

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

MathSciNet review:
776193

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Abstract: Let be a noncompact convex subset of a normed space . It is shown that if is not totally-bounded then there exists a Lipschitz self map of with , while if is totally-bounded then such a map does not exist, but still 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.

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

Article copyright:
© Copyright 1985
American Mathematical Society