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A compact convex set not convexly totally bounded. (English) Zbl 1015.46003
In connection with Schauder’s conjecture “Does every compact convex set in an arbitrary Hausdorff topological linear space have the fixed point property?”, A. Idzik [ibid. 35, 461-464 (1987; Zbl 0663.47036)] asked whether each compact convex set \(K\) in a Hausdorff topological linear space is convexly totally bounded, i.e., for each neighbourhood \(U\) of zero there are \(x_1,\dotsc,x_n\in K\) and convex subsets \(C_1,\dots,C_n\subset U\) such that \(K\subset\bigcup_{i=1}^n(x_i+C_i)\). A positive answer to Idzik’s question would have solved Schauder’s problem. The present author gives a simple example of a compact convex set in Ribe space [cf. M. Ribe, Proc. Am. Math. Soc. 237, 351-355 (1979; Zbl 0397.46002)] which is not convexly totally bounded. {In the meantime, R. Cauty [Fund. Math. 170, 231-246 (2001; Zbl 0983.54045)] has solved Schauder’s problem in the positive}.

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46A50 Compactness in topological linear spaces; angelic spaces, etc.
47H10 Fixed-point theorems