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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?

MSC:
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
46A55 Convex sets in topological linear spaces; Choquet theory
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References:
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