| Title:
|
Best approximations and porous sets (English) |
| Author:
|
Reich, Simeon |
| Author:
|
Zaslavski, Alexander J. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
681-689 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Banach space |
| Keyword:
|
complete metric space |
| Keyword:
|
generic property |
| Keyword:
|
Hausdorff metric |
| Keyword:
|
nearest point |
| Keyword:
|
porous set |
| MSC:
|
41A50 |
| MSC:
|
41A52 |
| MSC:
|
41A65 |
| MSC:
|
49K40 |
| MSC:
|
54E35 |
| MSC:
|
54E50 |
| MSC:
|
54E52 |
| idZBL:
|
Zbl 1096.41022 |
| idMR:
|
MR2062884 |
| . |
| Date available:
|
2009-01-08T19:32:09Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119422 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[9] Reich S., Zaslavski A.J.: Well-posedness and porosity in best approximation problems.Topol. Methods Nonlinear Anal. 18 395-408 (2001). Zbl 1005.41011, MR 1911709 |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
