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Keywords:
Banach space; complete metric space; generic property; Hausdorff metric; nearest point; porous set
Banach space; complete metric space; generic property; Hausdorff metric; nearest point; porous set
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 $\Cal F \subset S(X)$ 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.
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