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 PDF

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

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.

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