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Convexity of suns in tangent directions. (English) Zbl 1441.46012
Summary: A direction \(d\) is called a tangent direction to the unit sphere \(S\) if the conditions that \(s\in S\) and that \(\operatorname{aff}(s+d)\) is a tangent line to the sphere \(S\) at \(s\) imply that \(\operatorname{aff}(s+d)\) is a one-sided tangent to the sphere \(S\), i.e., it is the limit of secant lines at the point \(s\). A set \(M\) is called convex with respect to a direction \(d\) if \([x,y]\subset M\) whenever \(x,y\in M\), \((y-x)\parallel d\). It is shown that in an arbitrary normed space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to 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)
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