Strict Suns Composed of Planes

期刊论文
发表日期：

2024年12月1日

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A. R. Alimov

摘要

A set $M$ $𝑀$ is a strict sun if, for each $x\notin M$ $𝑥 𝑀$ , the set $P_{M}x$ $subscript 𝑃 𝑀 𝑥$ of best approximants from $M$ $𝑀$ for $x$ $𝑥$ is nonempty and each point $y\in P_{M}x$ $𝑦 subscript 𝑃 𝑀 𝑥$ is a nearest point from $M$ $𝑀$ for each point $z$ $𝑧$ from the ray emanating from $y$ $𝑦$ and passing through $x$ $𝑥$ . Strict suns are sometimes called Kolmogorov sets, because they satisfy the Kolmogorov criterion for best approximation. We study the structural properties of strict suns composed of a a finite number of planes (affine spaces, which may possibly degenerate to points). We always assume that the union of planes $M:=\bigcup L_{i}$ $assign 𝑀 subscript 𝐿 𝑖$ 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 $M:=\bigcup_{i=1}^{N}L_{i}$ $assign 𝑀 superscript subscript 𝑖 1 𝑁 subscript 𝐿 𝑖$ is a strict sun in a normed space, then $M$ $𝑀$ 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 $\ell^{\infty}_{n}$ $subscript superscript ℓ 𝑛$ . Namely, we show that if $M:=\bigcup_{i=1}^{N}L_{i}$ $assign 𝑀 superscript subscript 𝑖 1 𝑁 subscript 𝐿 𝑖$ is an irreducible union of planes in $\ell^{\infty}_{n}$ $subscript superscript ℓ 𝑛$ , $\Pi$ $Π$ is a bar (intersection of extreme hyperplanes), and $M\cap\Pi\neq\varnothing$ $𝑀 Π$ , then $M^{\prime}:=M\cap\Pi$ $assign superscript 𝑀 ′ 𝑀 Π$ is a strict sun in $\ell^{\infty}_{n}$ $subscript superscript ℓ 𝑛$ if and only if $M^{\prime}$ $superscript 𝑀 ′$ is convex, i.e., $M^{\prime}$ $superscript 𝑀 ′$ is the intersection of some plane $L_{i}$ $subscript 𝐿 𝑖$ with the bar $\Pi$ $Π$ . As a corollary, if $M:=\bigcup_{i=1}^{N}L_{i}$ $assign 𝑀 superscript subscript 𝑖 1 𝑁 subscript 𝐿 𝑖$ is a local strict sun in $\ell^{\infty}_{n}$ $subscript superscript ℓ 𝑛$ , then $M$ $𝑀$ consists of a single plane. Similar results are also established for sets $M:=\bigcup_{i=1}^{N}L_{i}$ $assign 𝑀 superscript subscript 𝑖 1 𝑁 subscript 𝐿 𝑖$ with continuous metric projection in $\ell^{\infty}_{n}$ $subscript superscript ℓ 𝑛$ . The present paper continues and develops the previous studies on approximation by Chebyshev sets composed of planes begun by A.R. Alimov and I.G. Tsar’kov in linear normed and asymmetrically normed spaces and the results of I.G. Tsar’kov on sets with a piecewise continuous metric projection.