Abstract
In this paper we establish a sufficient condition for the nonexistence of ƒ-antiproximinal sets (the notion of ƒ-antiproximinal set was introduced in [12] by the author together with T. Precupanu) in the space L 1 (S,∑,µ) of integrable functions on the measure space (S,∑,µ), when the function ƒ satisfies a certain special convexity property with respect to a measurable decomposition of the space S (Definition 2).
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
Bishop E. and Phelps R. - The support Junctionals of a convex set. Convexity Proc. Symp. Pure Math., VII, Amer. Math. Soc., 1963, 27–35.
Cobzaş S. -Antiproximinal sets in some Banach spaces, Math. Balkanica 4 (1974), 79–82.
Cobzaş S. -Convex antiproximinal sets in c 0 and c (Russian), Mat. Za- metclinki, 17 (1975), 449–457.
Cobzaş S. -Antiproximinal sets in Banach spaces of continuous functions, Revue d’Analyse Numérique et de Théorie de l’Approximation, 5 (1976), 127–143.
Edelstein M. - A note on nearest points, Quart. J. Math., Oxford 21 (1970), 403-405.
Edelstein M. and Thomson A.C. - Some results on nearest points and support properties of convex sets in c 0 , Pacific J. Math. 40 (1972), 553–560.
Floret K. - On the sum of two closed convex sets, Methods of Operations Research, 36 (1978), 73–85.
Holmes R. - A cours on optimization and best approximation, Lecture Notes in Math. No. 257, Springer Verlag, 1972, Berlin.
Holmes R. - Geometric Functional Analysis and Its Applications, Springer-Verlag, Berlin, 1975.
Klee V. - Remarks on nearest points in normed linear spaces, Proc. Col-loq. Convexity, Copenhagen, 1965, 168–176.
Precupanu A.–Analiză matematică. Funcţii reale, Edit. Did. Ped., Bu- cureşti, 1976.
Precupanu A. and Precupanu T. - A characterization of antiproximinal sets, An. Şt. Univ. Iaşi, 33 (1987), 99–105.
Singer I. - The Theory of the best approximation and Functional Analysis, Conf. Board of the Math. Sci. Soc. for Ind. and Applied Math., Philadelphia, Pensylvannia, 1974.
Stečkin S.B.–Caractérisation de I’approximation par des sous- ensembles d’espaces vectoriels normés, Rev. Mat. Pure et Appl. 8 (1963), 5–18.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Kluwer Academic Publishers
About this chapter
Cite this chapter
Precupanu, AM. (1998). A Convexity Condition for the Nonexistence of Some Antiproximinal Sets in the Space of Integrable Functions. In: Crouzeix, JP., Martinez-Legaz, JE., Volle, M. (eds) Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications, vol 27. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3341-8_8
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3341-8_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3343-2
Online ISBN: 978-1-4613-3341-8
eBook Packages: Springer Book Archive