Abstract
The concept of n-antiproximinal set in a Banach space is defined. The existence of convex closed n-antiproximinal sets in the spaces C and L 1 is studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. B. Holmes, A Course on Optimization and Best Approximation, Lect. Notes Math., Vol. 257 (Springer-Verlag, Berlin, 1972).
V. Klee, “Remarks on Nearest Points in Normed Linear Spaces,” in Proe. Golloq. Convexity. Copenhagen 1965 (Copenhagen, 1967), pp. 16H76.
I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Editura Academiei, Springer-Verlag, Bucharest, Berlin, 1970).
M. Edelstein, “A Note on Nearest Points,” Quart. J. Math. 21, 403 (1970).
M. Edelstein and A. C. Thompson, “Some Results on Nearest Points and Support Properties of Convex Sets in co,” Pacif. J. Math. 40 (3), 553 (1972).
S. Kobzash, “Convex Antiproximinal Sets in Spaces co and c,” Matem. Zametki 17 (3) 449, (1975). [Math. Notes 17 (3), 263 (1975)].
S. Cobza§, “Antiproximinal Sets in Banach Spaces of Continuous Functions,” Rev. anal, numer. §i teor. aproxim. 5, 127 (1976).
S. Cobza§, “Antiproximinal Sets in Banach Spaces of c 0-type,” Rev. anal, numer. §i teor. aproxim. 7, 141 (1978).
S. Cobza§, “Antiproximinal Sets in Some Banach Spaces,” Math. Balkanica 4, 79 (1974).
J. M. Borwein, “Some Remarks on a Paper of S. Cobzas on Antiproximinal Sets,” Bull. Calcutta Math. Soc. 73, 5 (1981).
M. Edelstein, “Weakly Proximinal Sets,” J. Approx. Theory 18 (1), 1 (1976).
R. R. Phelps, “Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space,” Can. Math. Bull. 31 (1), 121 (1988).
K. Floret, “On the Sum of Two Closed Convex Sets,” Meth. Oper. Res. 36, 73 (1978).
V. P. Fonf, “Antiproximinal Sets in Spaces of Continuous Functions on Compacta,” Matem. Zametki 33 (3), 549 (1983) [Math. Notes 33 (4), 282 (1983)].
V. S. Balaganskii, “Antiproximinal Sets in Spaces of Continuous Functions,” Matem. Zametki 60 (5), 643 (1996) [Math. Notes 60 (5), 485 (1996)].
V. S. Balaganskii, “On Antiproximinal Sets in Grothendieck Spaces,” Trudy Inst. Matem. i Mekhan. UrO RAN 18 (4), 90 (2012).
J. M. Borwein, Jimenez-Sevilla, and J. P. Moreno, “Antiproximinal Norms in Banach Spaces,” J. Approx. Theory 114, 57 (2002).
S. Cobza§, “Antiproximinal Sets in Banach Spaces,” Acta Univ. carol, math, et phys. 40 (2), 43 (1999).
P. A. Borodin, “On the Convexity of W-Chebyshev Sets,” Izvestiya RAN, Ser. Matem. 75 (5), 19 (2011) [Izvestiya: Math. 75 (5), 889 (2011)].
G. Sh. Rubinstein, “On an Extremal Problem in a Linear Normed Space,” Sib. Matem. Zh. VI (3), 711 (1965).
J. Diestel, Geometry of Banach Spaces, Lecture Notes in Math., Vol. 485 (Springer, Berlin, N.Y., 1975; Vysha Shkola, Kiev, 1980).
N. Dunford and J. T. Schwartz, Linear Operators, General Theory (John Wiley and Sons, Hoboken, NJ, 1988; IL, Moscow, 1962).
B. B. Bednov, “Steiner Points in the Space of Continuous Functions,” Vestnik Mosk. Univ., Matem. Mekhan., No. 6, 26 (2011) [Moscow Univ. Math. Bull. 66 (6), 255 (2011)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © B.B. Bednov, 2015, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2015, Vol. 70, No. 3, pp. 29–34.
About this article
Cite this article
Bednov, B.B. The n-antiproximinal sets. Moscow Univ. Math. Bull. 70, 130–135 (2015). https://doi.org/10.3103/S0027132215030067
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027132215030067