## 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

Zbl 0197.38601
Full Text:

### 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
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