Geometric construction of Chebyshev sets and suns in three-dimensional spaces with cylindrical norm

A geometric characterization of Chebyshev sets and suns in three-dimensional polyhedral spaces with cylindrical norm is presented. A number of new properties of Chebyshev sets, suns, and sets with continuous metric projection in three-dimensional cylindrical spaces is put forward. The new recent fact due to A. R. Alimov and E. V. Shchepin that suns and Chebyshev sets are convex in tangent directions to the unit sphere plays an important role in the paper.