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.