Best approximations and porous sets

Simeon Reich; Alexander J. Zaslavski

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 681-689
  • ISSN: 0010-2628

Abstract

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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 ℱ ⊂ S ( X ) such that its complement S ( X ) ∖ ℱ is σ -porous and such that for each A ∈ ℱ and each x ˜ ∈ D , the set of solutions of the best approximation problem ∥ x ˜ - z ∥ → min , z ∈ A , is nonempty and compact, and each minimizing sequence has a convergent subsequence.

How to cite

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Reich, Simeon, and Zaslavski, Alexander J.. "Best approximations and porous sets." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 681-689. <http://eudml.org/doc/249166>.

@article{Reich2003,
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 $\mathcal \{F\} \subset S(X)$ such that its complement $S(X) \setminus \mathcal \{F\}$ is $\sigma $-porous and such that for each $A\in \mathcal \{F\}$ and each $\tilde\{x\}\in D$, the set of solutions of the best approximation problem $\Vert \tilde\{x\}-z\Vert \rightarrow \min $, $z \in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.},
author = {Reich, Simeon, Zaslavski, Alexander J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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},
language = {eng},
number = {4},
pages = {681-689},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Best approximations and porous sets},
url = {http://eudml.org/doc/249166},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Reich, Simeon
AU - Zaslavski, Alexander J.
TI - Best approximations and porous sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 681
EP - 689
AB - 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 $\mathcal {F} \subset S(X)$ such that its complement $S(X) \setminus \mathcal {F}$ is $\sigma $-porous and such that for each $A\in \mathcal {F}$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\Vert \tilde{x}-z\Vert \rightarrow \min $, $z \in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.
LA - eng
KW - 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
UR - http://eudml.org/doc/249166
ER -

References

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  10. Reich S., Zaslavski A.J., A porosity result in best approximation theory, J. Nonlinear Convex Anal. 4 165-173 (2003). (2003) Zbl1024.41017MR1986978
  11. L. Zajíček, On the Fréchet differentiability of distance functions, Suppl. Rend. Circ. Mat. Palermo (2) 5 161-165 (1984). (1984) MR0781948
  12. Zajíček L., Porosity and σ -porosity, Real Anal. Exchange 13 314-350 (1987). (1987) MR0943561
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  14. Zaslavski A.J., Well-posedness and porosity in optimal control without convexity assumptions, Calc. Var. 13 265-293 (2001). (2001) Zbl1032.49035MR1864999

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