Abstract
The paper is concerned with problems of existence, uniqueness, and stability of best approximants and related problems of solarity for not necessarily closed sets (in particular, such sets are not necessarily proximinal). We give definitions of the projection boundary and of a projection closed set. We show that if \(X\) is a symmetrizable asymmetric Efimov–Stechkin space, then the set of points of approximative compactness of any nonempty projection closed set \(M\subset X\) is of second category in the metric exterior \(\operatorname{out}(M)\) of the set \(M\) . We also study relations between classes of sets in asymmetric Efimov–Stechkin spaces. We show that if \(X\) is a symmetrizable Efimov–Stechkin space and \(M\subset X\) is projection closed and \(\ell_0P_0\) -connected, then \(M\) is \(\ell_0\mathring B\) -connected. The solarity problem for sets of uniqueness is studied. In particular, the well-known theorem dating back to V. I. Berdyshev and V. L. Klee on solarity of boundedly compact Chebyshev sets is extended to the case of sets of uniqueness. Results on solarity of boundedly precompact projection closed sets of uniqueness are obtained. We also obtain results on preservation of solarity and other approximative properties of sets when changing to closed neighborhoods \(M+B(x,r)\) of sets. Special attention is given to the case of Chebyshev suns, for which a characterization theorem is obtained.
