
Journal of Convex Analysis 26 (2019), No. 4, [final page numbers not yet available] Copyright Heldermann Verlag 2019 Convexity of Suns in Tangent Directions Alexey R. Alimov Steklov Math. Institute, Russian Academy of Sciences, Moscow, Russia alexey.alimovmsu@yandex.ru Evgeny V. Shchepin Steklov Math. Institute, Russian Academy of Sciences, Moscow, Russia scepin@mi.ras.ru [Abstractpdf] 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~onesided 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 $x,y\in M$, $(yx)\parallel d$. 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 [ Fulltextpdf (86 KB)] for subscribers only. 