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Points of symmetry of convex sets in the two-dimensional complex space – a counterexample to D. Yost’s problem. (English) Zbl 0757.46017
The 3-ball intersection property ($$M$$-ideal property) was known not to be equivalent to the 2-ball intersection property for real spaces. In the complex case the answer was not known. By a construction of D. Yost, the latter problem was known to be equivalent to the question whether only compact convex sets $$K$$ in $$\mathbb{C}^ 2$$ which have a center of symmetry have the property that $$f(K)$$ is a disc for every linear map $$f:\mathbb{C}^ 2\to\mathbb{C}$$.
The author constructs a nice counterexample: There is a set of the form $$K=\{(z,w)\mid\;| z|\leq 1$$, $$| w|\leq\varphi(z)\}$$ such that $$f(K)$$ is a disc for every linear $$f:\mathbb{C}^ 2\to\mathbb{C}$$, but has no center of symmetry. Consequently, the 3- and 2-ball properties differ in the complex case, too.
Reviewer: H.König (Kiel)

##### MSC:
 46B20 Geometry and structure of normed linear spaces
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##### References:
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