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The geometric structure of Chebyshev sets in \(\ell^\infty(n)\). (English. Russian original) Zbl 1087.41027

Funct. Anal. Appl. 39, No. 1, 1-8 (2005); translation from Funkts. Anal. Prilozh. 39, No. 1, 1-10 (2005).
Let \(M\) (resp.\(H\)) be a set (resp. subspace) in \(R^n\). Some approximative properties of the set \(M\cap N \) in the subspace \(H\), where the norm on \(H\) is induced from \(l^\infty (n) \), is studied. The results for coordinate and noncoordinate subspaces \(H\) are completely different. There are two main results in the paper. Theorem 1 describes the approximative properties of intersections of Chebyshev sets, suns, and strict suns in \(l^\infty (n) \) with coordinate subspaces. Theorem 2 characterizes Chebyshev sets in \(l^\infty (n) \) in geometric terms.

MSC:

41A50 Best approximation, Chebyshev systems
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