A compact convex set not convexly totally bounded (Q2770173)

In connection with Schauder's conjecture ``Does every compact convex set in an arbitrary Hausdorff topological linear space have the fixed point property?'', \textit{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. \textit{M. Ribe}, Proc. Am. Math. Soc. 237, 351-355 (1979; Zbl 0397.46002)] which is not convexly totally bounded. \{In the meantime, \textit{R. Cauty} [Fund. Math. 170, 231-246 (2001; Zbl 0983.54045)] has solved Schauder's problem in the positive\}.