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

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


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