Chebyshev sets that are unions of planes (Q6639444)

This paper is on a Chebyshev set that is the union of at most countably many planes and is irreducible.\N\NA Chebyshev set is a set such that each of its points has a unique best approximation in this set. \textit{A. R. Alimov} and \textit{I. G. Tsar'kov} [J. Approx. Theory 298, Article ID 106009, 12 p. (2024; Zbl 1544.46010)] studied Chebyshev sets composed of at most countably many approximately compact sets and showed that a Chebyshev set in a uniformly smooth Banach space is a Cheyshev sun. This paper continues and generalizes results in [\textit{A. R. Alimov} and \textit{I. G. Tsar'kov}, J. Approx. Theory 298, Article ID 106009, 12 p. (2024; Zbl 1544.46010)] and [\textit{I. G. Tsar'kov}, Math. Notes 113, No. 6, 840--849 (2023; Zbl 1520.41008); translation from Mat. Zametki 113, No. 6, 905--917 (2023)].\N\NIt states that the following conditions may conflict with each other (in the sense that they can not hold simultaneously).\N\begin{itemize}\N\item[(1)] \(M\) is an at most countable irreducible union of planes in an asymmetric normed space \(X\) with some restrictions.\N\N\item[(2)] \(M\) is \(B\)-connected or \(\mathring{B}\)-connected or regularly right-approximately compact or unimodal.\N\N\item[(3)] \(M\) is a Chebyshev set.\N\end{itemize}\N\NMoreover, the paper states that in a symmetrizable asymmetric CLUR-space that is reflexive, that is, an Efimov-Stechkin space, a set that is at most countable irreducible union of planes can not be a Chebyshev set.\N\NFurthermore, the paper states in an asymmetric normed space, the finite union of planes is a Chebyshev set if and only if it is a Chebyshev plane.\N\NThe results of the paper can lead to the result that in \(L^1(\Omega,\sigma, \mu)\), at most countable irreducible union of planes of finite dimension or co-dimension is never a Chebyshev set.