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Best approximations and porous sets. (English) Zbl 1096.41022

Summary: Let \(D\) be a nonempty compact subset of a Banach space \(X\) and denote by \(S(X)\) the family of all nonempty bounded closed convex subsets of \(X\). We endow \(S(X)\) with the Hausdorff metric and show that there exists a set \(\mathcal F \subset S(X)\) such that its complement \(S(X) \setminus \mathcal F\) is \(\sigma \)-porous and such that for each \(A\in \mathcal F\) and each \(\tilde x\in D\), the set of solutions of the best approximation problem \(\| \tilde x-z\| \to \min ,\;z \in A\), is nonempty and compact, and each minimizing sequence has a convergent subsequence.

MSC:

41A50 Best approximation, Chebyshev systems
41A52 Uniqueness of best approximation
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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