Abstract:
A set is called a Chebyshev set if it contains a unique best approximation element. We study the structure of the
complements of Chebyshev sets, in particular considering the following question: How many connected components can the complement of a Chebyshev set in a finite-dimensional normed or nonsymmetrically normed linear space have? We extend some results from [A. R. Alimov, East J. Approx, 2, No. 2, 215–232 (1996)]. A. L. Brown's characterization of four-dimensional normed linear spaces in which every Chebyshev set is convex is extended to the nonsymmetric setting. A characterization of finite-dimensional spaces that contain a strict sun whose complement has a given number of connected components is established.
Citation:
A. R. Alimov, “On the Structure of the Complements of Chebyshev Sets”, Funktsional. Anal. i Prilozhen., 35:3 (2001), 19–27; Funct. Anal. Appl., 35:3 (2001), 176–182
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\paper On the Structure of the Complements of Chebyshev Sets
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\vol 35
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\pages 19--27
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\jour Funct. Anal. Appl.
\yr 2001
\vol 35
\issue 3
\pages 176--182
\crossref{https://doi.org/10.1023/A:1012370610709}
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Linking options:
https://www.mathnet.ru/eng/faa255
https://doi.org/10.4213/faa255
https://www.mathnet.ru/eng/faa/v35/i3/p19
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