
Journal of Convex Analysis 26 (2019), No. 4, 10711076 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 (91 KB)] for subscribers only. 