A compact convex set not convexly totally bounded. (English) Zbl 1015.46003

Trombetta, Giulio

Bull. Pol. Acad. Sci., Math. 49, No. 3, 223-228 (2001).

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}.

Reviewer: Christian Fenske (Gießen)

MSC:

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

Keywords:

convex set; convexly totally bounded set; needle point; Ribe space; Schauder’s fixed point theorem

Citations:

Zbl 0663.47036; Zbl 0397.46002; Zbl 0983.54045