Geodesic
Chebyshev sets composed of subspaces in asymmetric normed spaces
Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1032-1049

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By definition, a Chebyshev set is a set of existence and uniqueness, that is, any point has a unique best approximant from this set. We study properties of Chebyshev sets composed of finitely or infinitely many planes (closed affine subspaces, possibly degenerated to points). We show that a finite union of planes is a Chebyshev set if and only if is a Chebyshev plane. Under some conditions on a space or a set, we show that a countable union of planes is never a Chebyshev set (unless this union is a Chebyshev plane itself). As a corollary, we give the following partial answer to the famous Efimov–Stechkin–Klee problem on convexity of Chebyshev sets: in Hilbert spaces (and, more generally, in reflexive (CLUR)-spaces), an at most countable union of planes is a Chebyshev set if and only if this set is a Chebyshev plane. Results of this kind are obtained both in usual normed linear spaces and in spaces with asymmetric norm.
Keywords: Chebyshev set, best approximation, union of subspaces, asymmetric normed space, sun, ridge function. 
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     title = {Chebyshev sets composed of subspaces in asymmetric normed spaces},
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A. R. Alimov; I. G. Tsar'kov. Chebyshev sets composed of subspaces in asymmetric normed spaces. Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1032-1049. http://geodesic.mathdoc.fr/item/IM2_2024_88_6_a1/