Max-Solar Properties of Sets in Normed and Asymmetrically Normed Spaces

The paper is concerned with new concepts of the max-approximation theory related to solar properties of sets: a max-sun, a local max-sun, max-Chebyshev set, and a max-attractor. Some results on max-solarity are established for uniquely remotal sets (such sets are called absolute max-Chebyshev sets). An example of a nonsingleton (nonclosed) uniquely remotal set on an asymmetrically normed plane is constructed, which gives a negative answer in the problem of whether a uniquely remotal set is a singleton (the unique farthest point problem). Results are obtained both in symmetrically and asymmetrically normed spaces.