Connectedness and other geometric properties of suns and Chebyshev sets

This survey is concerned with structural characteristics of “suns” in normed linear spaces. Special attention is paid to connectedness and monotone path-connectedness of suns. We address both direct theorems of the geometric approximation theory, in which approximative properties of sets are derived from their structural characteristics, and inverse theorems, in which from approximative characteristics of sets one derives their structural properties.