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, $M\subset X$ is projection closed, and $\ell_0P_0$-connected, then $M$ is $\ell_0\overset{\ \circ}{B}$-connected. 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.
Keywords: projection boundary, projection closed set, connected set, best approximation, sun, Chebyshev sun, Efimov–Stechkin space, metrical exterior of a set, approximative compactness, approximative precompactness
Received April 1, 2025
Revised June 6, 2025
Accepted June 9, 2025
Funding Agency: The paper was published with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement no. 075-15-2025-013.
Alexey R. Alimov, Dr. Phys.-Math. Sci., Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119899 Russia; St. Petersburg State University, St. Petersburg, 199034 Russia, e-mail: alexey.alimov-msu@yandex.ru
Igor’ G. Tsar’kov, Prof., Dr. Phys.-Math. Sci., Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119899 Russia; Moscow Center for Fundamental and Applied Mathematics, Moscow, 119899 Russia, e-mail: tsar@mech.math.msu.su
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Cite this article as: A.R. Alimov, I.G. Tsar’kov. Projection closed sets and Efimov–Stechkin spaces. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 3, pp. 20–35.
