Best approximation in metric spaces. (English) Zbl 0652.51019

Khalil, Roshdi

Proc. Am. Math. Soc. 103, No. 2, 579-586 (1988).

A metric space (X,d) is called an M-space if for every x and y in X and for every \(r\in [0,\lambda]\), where \(\lambda =d(x,y)\), there is a \(z\in X\) such that \(B[x,r]\cap B[y,\lambda -r]=\{z\}.\) Also, if G is a closed subset of X and \(\rho (x,G)=\inf \{d(x,y):\) \(y\in G\}\), then G is called proximinal in X if the infimum is attained for all \(x\in X\). The author studies M-spaces in terms of proximinality properties of certain sets.

Reviewer: E.J.F.Primrose

MSC:

51K05	General theory of distance geometry
51K99	Distance geometry
41A50	Best approximation, Chebyshev systems

Keywords:

M-spaces; proximinality

References:

[1]	G. C. Ahuja, T. D. Narang, and Swaran Trehan, Best approximation on convex sets in metric linear spaces, Math. Nachr. 78 (1977), 125 – 130. · Zbl 0292.41023 · doi:10.1002/mana.19770780110
[2]	Günter Albinus, Approximation in metric linear spaces, Approximation theory (Papers, VIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975) Banach Center Publ., vol. 4, PWN, Warsaw, 1979, pp. 7 – 18.
[3]	N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405 – 439. · Zbl 0074.17802
[4]	Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. · Zbl 0050.38502
[5]	Anthony D. Berard Jr., Characterizations of metric spaces by the use of their midsets: Intervals, Fund. Math. 73 (1971/72), no. 1, 1 – 7. · Zbl 0223.54017
[6]	Herbert Busemann, Metric Methods in Finsler Spaces and in the Foundations of Geometry, Annals of Mathematics Studies, no. 8, Princeton University Press, Princeton, N. J., 1942. · Zbl 0063.00672
[7]	Herbert Busemann, Note on a theorem on convex sets, Mat. Tidsskr. B. 1947 (1947), 32 – 34. · Zbl 0040.38403
[8]	Saichi Izumino, Khalil’s theorem and a property of uniformly convex spaces, Math. Rep. Toyama Univ. 6 (1983), 41 – 46. · Zbl 0524.46009
[9]	Roshdi Khalil, Extreme points of the unit ball of Banach spaces, Math. Rep. Toyama Univ. 4 (1981), 41 – 45. · Zbl 0473.46012
[10]	Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Translated from the Romanian by Radu Georgescu. Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. · Zbl 0197.38601
[11]	Karl Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 100 (1928), no. 1, 75 – 163 (German). · JFM 54.0622.02 · doi:10.1007/BF01448840