Chebyshev sets composed of subspaces in asymmetric normed spaces

Abstract

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.