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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)
##### Keywords:
tangent direction; sun; Chebyshev set
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##### References:
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