Monotone path-connectedness of Chebyshev sets in three-dimensional spaces

Monotone path-connectedness of Chebyshev sets in three-dimensional spaces

We characterize the three-dimensional Banach spaces in which any Chebyshev set is monotone path-connected. Namely, we show that in a three-dimensional space each Chebyshev set is monotone path- connected if and only if one of the following two conditions is satisfied: any exposed point of the unit s...

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Veröffentlicht in:Sbornik. Mathematics 2021-05, Vol.212 (5), p.636-654
Hauptverfasser: Alimov, A. R., Bednov, B. B.
Format: Artikel
Sprache:eng
Schlagworte:
Banach spaces
Chebyshev approximation
Chebyshev set
cylindrical norm
monotone path-connected set
Set theory
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Zusammenfassung:We characterize the three-dimensional Banach spaces in which any Chebyshev set is monotone path-connected. Namely, we show that in a three-dimensional space each Chebyshev set is monotone path- connected if and only if one of the following two conditions is satisfied: any exposed point of the unit sphere of is a smooth point or (that is, the unit sphere of is a cylinder). Bibliography: 17 titles.
ISSN:1064-5616
1468-4802
DOI:10.1070/SM9325