Best approximations and porous sets
Reich, Simeon ; Zaslavski, Alexander J.
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003), p. 681-689 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 2003-01-01
Classification:  41A50,  41A52,  41A65,  49K40,  54E35,  54E50,  54E52 
@article{119422,
     author = {Simeon Reich and Alexander J. Zaslavski},
     title = {Best approximations and porous sets},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {44},
     year = {2003},
     pages = {681-689},
     zbl = {1096.41022},
     mrnumber = {2062884},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119422}
}

Reich, Simeon; Zaslavski, Alexander J. Best approximations and porous sets. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 681-689. http://gdmltest.u-ga.fr/item/119422/

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