A result on best proximity pair of two sets. (English) Zbl 0653.41030

Xu, Xiubin

J. Approximation Theory 54, No. 3, 322-325 (1988).

Let F and G be nonvoid subsets of the normed space X. The pair \((f_ 0,g_ 0)\in F\times G\) is called a best proximity pair of F and G if: \(\| f_ 0-g_ 0\| =\inf \{\| f-g\|:\quad f\in F,\quad g\in G\}.\) The author’s main result is: If F is sequentially compact, G is convex and every \(f\in F\) has a best approximation element in G then a best proximity pair of F and G exists. A variant of this proposition for pairs of sets in a metric space is also considered.

Reviewer: I.Şerb

MSC:

41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50	Best approximation, Chebyshev systems

Keywords:

best proximity pair

References:

[1]	Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (1970), Springer-Verlag: Springer-Verlag New York · Zbl 0197.38601
[2]	Pai, D. V., Proximal points of convex sets in normed linear spaces, Yokohama Math. J., 22, 53-78 (1974) · Zbl 0295.41017
[3]	Sahney, B. N.; Singh, S. P., On best simultaneous approximation, (Cheney, E. W., Approximation Theory III (1980)), 783-789, New York · Zbl 0492.41028
[4]	Xu, Xiubin, On best proximity pairs and best mutual approximations, J. Zhejiang Normal Univ., 6, 49-54 (1983)