Convexity of Suns in Tangent Directions

Abstract

A direction $$d$$ is called a tangent direction to the unit sphere $$S$$ if the conditions that $$s \in S$$ and $${\text{lin}}(s + d)$$ is a supporting line to $$S$$ at the point s imply that $${\text{lin}}(s + d)$$ is a semitangent line to S, i.e., is the limit of secants at s. A set M is called convex in a direction $$d$$ if $$x,y \in M$$ and $$(y - x)\parallel d$$ imply that $$[x,y] \subset M$$ . In an arbitrary normed linear space, an arbitrary sun (in particular, a boundedly compact Chebyshev set) is proved to be convex in any tangent direction of the unit sphere.