Journal of Convex Analysis 26 (2019), No. 4, 1071--1076
Copyright Heldermann Verlag 2019
Convexity of Suns in Tangent Directions
Alexey R. Alimov
Steklov Math. Institute, Russian Academy of Sciences, Moscow, Russia
alexey.alimov-msu@yandex.ru
Evgeny V. Shchepin
Steklov Math. Institute, Russian Academy of Sciences, Moscow, Russia
scepin@mi.ras.ru
[Abstract-pdf]
A direction $d$ is called a tangent direction to the unit sphere $S$ if the conditions $s\in S$ and $\operatorname{aff}(s+d)$ 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.
Keywords: Sun, Chebyshev set, directional convexity.
MSC: 41A65, 52A05
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