Convexity of Suns in Tangent Directions

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.