Suns are convex in tangent directions

A direction d is called a tangent direction to the unit sphere S of a normed linear space s  S and lin(s + d) is a tangent line to the sphere S at s imply that lin(s + d) is a one-sided tangent to the sphere S , i. e., it is the limit of secant lines at s . A set M is called convex with respect to a direction d if [x, y]  M whenever x , y in M , ( y − x ) || d . We show that in a normed linear space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to any tangent direction of the unit sphere.