A set $M$ is a strict sun if, for each $x\notin M$, the set $P_Mx$ of best approximants from $M$ for $x$ is nonempty and each point $y\in P_Mx$ is a nearest point from $M$ for each point $z$ from the ray emanating from $y$ and passing through $x$. Strict suns are sometimes called Kolmogorov sets, because they satisfy the Kolmogorov criterion for best approximation. We study structural properties of strict suns composed of a finite number of planes (affine spaces, which may possibly degenerate to points). We always assume that the union of planes $M:=\bigcup L_i$ is irreducible, i.e., no plane in this union contains another plane from the union. We show that if an irreducible finite union of planes $M :=\bigcup_{i=1}^N L_i$ is a strict sun in a normed space, then $M$ consists of a single plane. In this result, the strict sun cannot be replaced by a sun. A stronger local analog of this result is proved in the space $\ell^\infty_n$. Namely, we show that if $M :=\bigcup_{i=1}^N L_i$ is an irreducible union of planes in $\ell^\infty_n$, $\Pi$ is a bar (intersection of extreme hyperplanes), and $M\cap \Pi\ne \emptyset$, then $M':=M\cap \Pi$ is a strict sun in $\ell^\infty_n$ if and only if $M'$ is convex, i.e., $M'$ is the intersection of some plane $L_i$ with the bar $\Pi$. As a corollary, if $M :=\bigcup_{i=1}^N L_i$ is a local strict sun in $\ell^\infty_n$, then $M$ consists of a single plane. Similar results are also established for sets $M :=\bigcup_{i=1}^N L_i$ with continuous metric projection in $\ell^\infty_n$. The present paper continues and develops the previous studies on approximation by Chebyshev sets composed of planes began by the author of the article and I.G. Tsar'kov in linear normed and asymmetrically normed spaces and the results of I.G. Tsar'kov on sets with piecewise continuous metric projection.
Keywords: best approximation, union of planes, sun, strict sun, discretization
Received June 23, 2024
Revised August 27, 2024
Accepted September 2, 2024
Funding Agency: This research was conducted at Lomonosov Moscow State University with the support of the Russian Science Foundation (project no. 22-11-00129).
Alexey R. Alimov, Dr. Phys.-Math. Sci., Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119899 Russia; Moscow Center for Fundamental and Applied Mathematics, Moscow, 119899 Russia, e-mail: alexey.alimov-msu@yandex.ru
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Cite this article as: A.R. Alimov. Strict suns composed of planes. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 27–36.
