Monotone path-connectedness of Chebyshev sets in three-dimensional spaces

Abstract

We characterize the three-dimensional Banach spaces in which any Chebyshev set is monotone path-connected. Namely, we show that in a three-dimensional space each Chebyshev set is monotone path- connected if and only if one of the following two conditions is satisfied: any exposed point of the unit sphere of is a smooth point or (that is, the unit sphere of is a cylinder). Bibliography: 17 titles.