Reich, Simeon; Zaslavski, Alexander J. Best approximations and porous sets. (English) Zbl 1096.41022 Commentat. Math. Univ. Carol. 44, No. 4, 681-689 (2003). 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. Cited in 4 Documents MSC: 41A50 Best approximation, Chebyshev systems 41A52 Uniqueness of best approximation 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) Keywords:Banach space; complete metric space; generic property; Hausdorff metric; nearest point; porous set PDFBibTeX XMLCite \textit{S. Reich} and \textit{A. J. Zaslavski}, Commentat. Math. Univ. Carol. 44, No. 4, 681--689 (2003; Zbl 1096.41022) Full Text: EuDML EMIS