×

Characterizations of simultaneous farthest point in normed linear spaces with applications. (English) Zbl 1154.90620

Summary: In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space \(X\). We study simultaneous farthest point in \(X\) by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball \({B_{X^{*}}}\) of \(X^{*}\), where \(X^{*}\) is the dual space of \(X\). We establish the characterizations of simultaneous farthest points for bounded sets in \({C_{\mathbb{R}}(Q)}\), the space of all real-valued continuous functions on a compact topological space \(Q\) endowed with the usual operations and with the norm \({\| x \|=\max_{q\in Q}\mid x(q) \mid}\). It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Bucharest Academy of the Socialist Republic of Romania; Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0197.38601), Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in \(X\) which specifically concentrates on the Hahn-Banach Theorem in \(X\).

MSC:

90C48 Programming in abstract spaces

Citations:

Zbl 0197.38601
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baronti M.: A note on remotal sets in Banach spaces. Publications De L’Institut Mathematique, Nouvelle serie tome 53(67), 95–98 (1993) · Zbl 0809.46012
[2] Borwein J.M.: Proximality and Chebyshev sets. J. Optim. Lett. 1, 21–32 (2007) · Zbl 1138.46009
[3] Bosznay A.P.: A remark on uniquely remotal sets in C(K,X). Period. Math. Hungar. 12(1), 11–14 (1981) · Zbl 0449.46024
[4] Elumalai S., Vijayaragavan R.: Farthest points in normed linear spaces. J. Gen. Math. 14(3), 9–22 (2006) · Zbl 1164.41342
[5] Halmos, P.R.: A Course on Optimization and Best Approximation, Lecture Notes in Mathematics, vol. 257. Springer, Heidelberg (1972)
[6] Maaden A.: On the C-farthest points. Extracta Math. 16(2), 211–222 (2001) · Zbl 1010.46012
[7] Narang T.D.: A result on Chebyshev centers. Mat. Vesnik 38, 197–198 (1986) · Zbl 0613.41024
[8] Niknam A.: On uniquely remotal sets. Indian J. Pure Appl. Math. 15(10), 1079–1083 (1984)
[9] Singer I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, Berlin (1970) · Zbl 0197.38601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.