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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.

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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References:
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