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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.
MSC:
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)
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