Geodesic
Best approximations and porous sets
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 681-689.

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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 $\Cal F \subset S(X)$ such that its complement $S(X) \setminus \Cal F$ is $\sigma$-porous and such that for each $A\in \Cal F$ 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.
Classification : 41A50, 41A52, 41A65, 49K40, 54E35, 54E50, 54E52
Keywords: Banach space; complete metric space; generic property; Hausdorff metric; nearest point; porous set 
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     title = {Best approximations and porous sets},
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Reich, Simeon; Zaslavski, Alexander J. Best approximations and porous sets. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 681-689. http://geodesic.mathdoc.fr/item/CMUC_2003__44_4_a11/