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Convexly totally bounded and strongly totally bounded sets. Solution of a problem of Idzik. (English) Zbl 0805.47055
A. Idzik [Bull. Pol. Acad. Sci., Math. 35, 461-464 (1987; Zbl 0663.47036)] introduced the concept of convexly totally bounded subsets of a topological linear space and proved that every convexly totally bounded set has the fixed point property. Consequently, the Schauder conjecture (see the preceding review), would be proved if one could show that every compact convex set is convexly totally bounded. In this paper the authors give an example of a compact convex set in \(L_ p[0,1]\) \((p<1)\) which is not convexly totally bounded. Moreover, they introduce and study a “measure of non-convexly total boundedness”.

MSC:
47H10 Fixed-point theorems
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