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Convexity of suns in tangent directions. (English. Russian original) Zbl 1427.46008
Dokl. Math. 99, No. 1, 14-15 (2019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 484, No. 2, 131-133 (2019).
Summary: A direction \(d\) is called a tangent direction to the unit sphere \(S\) if the conditions that \(s \in S\) and \(\operatorname{lin}(s + d)\) is a supporting line to \(S\) at the point \(s\) imply that \(\operatorname{lin}(s + d)\) is a semitangent line to \(S\), i.e., is the limit of secants at \(s\). A set \(M\) is called convex in a direction \(d\) if \(x,y \in M\) and \((y - x)\parallel d\) imply that \([x,y] \subset M\). In an arbitrary normed linear space, an arbitrary sun (in particular, a boundedly compact Chebyshev set) is proved to be convex in any tangent direction of the unit sphere.
46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI
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