Simeon Reich, Alexander J. Zaslavski
Best approximations and porous sets

Comment.Math.Univ.Carolinae 44,4 (2003) 681-689.

Abstract: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.

Keywords: Banach space, complete metric space, generic property, Hausdorff metric, nearest point, porous set
AMS Subject Classification: 41A50, 41A52, 41A65, 54E35, 54E50, 54E52