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On typical compact convex sets in Hilbert spaces. (English) Zbl 0976.46010

Summary: Let \(\mathbb{E}\) be an infinite-dimensional separable space and for \(e\in\mathbb{E}\) and \(X\) a nonempty compact convex subset of \(\mathbb{E}\), let \(q_X(e)\) be the metric antiprojection of \(e\) on \(X\). Let \(n\geq 2\) be an arbitrary integer. It is shown that for a typical (in the sense of the Baire category) compact convex set \(X\subset\mathbb{E}\) the metric antiprojection \(q_X(e)\) has cardinality at least \(n\) for every \(e\) in a dense subset of \(\mathbb{E}\).

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B20 Geometry and structure of normed linear spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
54E52 Baire category, Baire spaces
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