Abstract
Given a nonempty closed bounded subset X of a Banach space E, we study the ambiguous locus A e (X) (resp. A u (X), A wp (X)) of X i.e. the set of all points z ∈ E such that the nearest point mapping from X to z fails to have existence (resp. uniqueness, well posedness). If E is uniformly convex, we show that A wp (X) is σ-porous in E. Moreover, we prove that for most (in the sense of the Baire category) nonempty closed bounded subsets X of E the set A wp (X) is uncountable and dense in E, provided that E is separable, strictly convex, and of dimension ≥ 2. Finally, under the same assumptions on E, we prove that for most nonempty compact convex subsets X of E the ambiguous locus A u (X), relative to the farthest point mapping, is uncountable and dense in E
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Asplund, E. (1966), Farthest points in reflexive locally uniformly rotund Banach spaces, Israel J. Math. 4, pp. 213–216.
Bartke, K. and Berens, H. (1986), Eine Beschreibung der Nichteindentigkeitsmenge für die beste Approximation in der Euklidischen Ebene, J. Approx. Theory 47, pp. 54–74.
Borwein, J.M. and Fitzpatrick, S. (1989), Existence of nearest points in Banach spaces, Can. J. Math. 41, pp. 702–720.
De Blasi, F.S. and Myjak, J. Ambiguous loci of the nearest point mapping in Banach spaces, (submitted).
De Blasi, F.S. and Myjak, J. Some typical properties of compact sets in Banach spaces, (submitted).
De Blasi, F.S. and Myjak, J. Ambiguous loci of the farthest point mapping from compact convex sets, (submitted).
De Blasi, F.S., Myjak, J., Papini, P.L. Porous sets in best approximation Theory, J. London Math. Soc. (to appear).
Dolzhenko, E. (1967) Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat., 31, pp. 3–14.
Dontchev, A. and Zolezzi, T. Well posed optimization problems, (to appear).
Furi, M. and Vignoli, A. (1970), About well posed optimization problems for functionals in metric spaces, J. Optim. Theory Appl., 5, pp. 225–229.
Edelstein, M. (1966), Farthest points of sets in uniformly convex Banach Spaces, Israel J. Math., 4, pp. 171–176.
Gruber, P. and Zamfirescu, T. (1990), Generic properties of compact starshaped, sets, Proc. Amer. Soc., 108, pp. 207–214.
Gruber, P. and Kenderov, P. (1982), Approximation of convex bodies by Polytopes, Rend. Circ. Mat. Palermo, 31, pp. 195–225.
Klee, V.L. (1959), Some new results on smoothness and rotundity in normed linear paces, Math. Ann., 139, pp. 51–63.
Ka-Sing Lau (1978), Almost Chebyshev subspaces in reflexive Banach spaces, Indiana Univ. Math. J., 27, pp. 791–795.
Konjagin, S.V. (1980), On approximation properties of closed sets in Banach spaces and the characterization of strongly convex spaces, Soviet Math. Dokl., 21, pp. 418–422.
Konjagin, S.V. (1983), Sets of points of nonemptyness and continuity of the metric projection, Matemat. Zametki, 33, pp. 331–338.
Stečkin, S. (1963), Approximative properties of subsets of Banach spaces (Russian), Rev. Roum. Math. Pures Appl., 8, pp. 5–8.
Zajíček, T. (1985), On the Fréchet differentiability of distance functions, Rend. Circ. Mat. Palermo (2)5, Supplemento, pp. 161–165.
Zajíček, L. (1987/88), Porosity and σ-porosity, Real Analysis Exchange, 13(2), pp. 314–350.
Zamfirescu, T. (1985), Using Baire category in geometry, Rend. Sem. Mat. Univers. Politecn. Torino 43, 1, pp. 67–88.
Zamfirescu, T. (1990), The nearest point mapping is single valued nearly Everywhere, Arch. Math. 51, pp. 563–566.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
De Blasi, F.S., Myjak, J. (1992). Ambiguous Loci in Best Approximation Theory. In: Singh, S.P. (eds) Approximation Theory, Spline Functions and Applications. NATO ASI Series, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2634-2_21
Download citation
DOI: https://doi.org/10.1007/978-94-011-2634-2_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5164-4
Online ISBN: 978-94-011-2634-2
eBook Packages: Springer Book Archive