Previous |  Up |  Next


Banach space; complete metric space; generic property; Hausdorff metric; nearest point; porous set
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.
[1] Benyamini Y., Lindenstrauss J.: Geometric Nonlinear Functional Analysis. Amer. Math. Soc. Providence, RI (2000). MR 1727673 | Zbl 0946.46002
[2] De Blasi F.S., Georgiev P.G., Myjak J.: On porous sets and best approximation theory. preprint. Zbl 1088.41015
[3] De Blasi F.S., Myjak J.: On the minimum distance theorem to a closed convex set in a Banach space. Bull. Acad. Polon. Sci. 29 373-376 (1981). MR 0640331 | Zbl 0515.41031
[4] De Blasi F.S., Myjak J.: On almost well posed problems in the theory of best approximation. Bull. Math. Soc. Sci. Math. R.S. Roum. 28 109-117 (1984). MR 0771542 | Zbl 0593.41026
[5] De Blasi F.S., Myjak J., Papini P.L.: Porous sets in best approximation theory. J. London Math. Soc. 44 135-142 (1991). MR 1122975 | Zbl 0786.41027
[6] Furi M., Vignoli A.: About well-posed minimization problems for functionals in metric spaces. J. Optim. Theory Appl. 5 225-229 (1970).
[7] Matoušková E.: How small are the sets where the metric projection fails to be continuous. Acta Univ. Carolin. Math. Phys. 33 99-108 (1992). MR 1287230
[8] Reich S., Zaslavski A.J.: Asymptotic behavior of dynamical systems with a convex Lyapunov function. J. Nonlinear Convex Anal. 1 107-113 (2000). MR 1751731 | Zbl 0984.37016
[9] Reich S., Zaslavski A.J.: Well-posedness and porosity in best approximation problems. Topol. Methods Nonlinear Anal. 18 395-408 (2001). MR 1911709 | Zbl 1005.41011
[10] Reich S., Zaslavski A.J.: A porosity result in best approximation theory. J. Nonlinear Convex Anal. 4 165-173 (2003). MR 1986978 | Zbl 1024.41017
[11] L. Zajíček: On the Fréchet differentiability of distance functions. Suppl. Rend. Circ. Mat. Palermo (2) 5 161-165 (1984). MR 0781948
[12] Zajíček L.: Porosity and $\sigma$-porosity. Real Anal. Exchange 13 314-350 (1987). MR 0943561
[13] Zajíček L.: Small non-$\sigma$-porous sets in topologically complete metric spaces. Colloq. Math. 77 293-304 (1998). MR 1628994
[14] Zaslavski A.J.: Well-posedness and porosity in optimal control without convexity assumptions. Calc. Var. 13 265-293 (2001). MR 1864999 | Zbl 1032.49035
Partner of
EuDML logo