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Best approximation in metric spaces. (English) Zbl 0652.51019
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
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