Journal of Convex Analysis 30 (2023), No. 1, 159--174
Copyright Heldermann Verlag 2023
Max-Solar Properties of Sets in Normed and Asymmetrically Normed Spaces
Alexey R. Alimov
Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
alexey.alimov-msu@yandex.ru
Igor' G. Tsar'kov
Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia
igtsarkov@yandex.ru
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.
Keywords: Max-approximation, farthest point, absolutely max-Chebyshev set, uniquely remotal set, max-sun, local max-sun, point of max-luminosity, Chebyshev center.
MSC: 41A65, 52A21, 41A28.
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