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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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
MathSciNet review: 776193
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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\Vert {x - f\left( x \right)} \right\Vert: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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0776193-5
PII: S 0002-9939(1985)0776193-5
Article copyright: © Copyright 1985 American Mathematical Society