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Проекционно замкнутые множества и пространства Ефимова – Стечкина. / Alimov, A.R.; Tsar’kov, I.G.
в: Trudy Instituta Matematiki i Mekhaniki UrO RAN, Том 31, № 3, 2025, стр. 20-35.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - Проекционно замкнутые множества и пространства Ефимова – Стечкина
AU - Alimov, A.R.
AU - Tsar’kov, I.G.
N1 - Export Date: 30 March 2026; Cited By: 0
PY - 2025
Y1 - 2025
N2 - 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 ⊂ X is of second category in the metric exterior 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 ⊂ X is projection closed, and ℓ0P0-connected, then M is ℓ0B˚-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. © 2025, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.
AB - 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 ⊂ X is of second category in the metric exterior 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 ⊂ X is projection closed, and ℓ0P0-connected, then M is ℓ0B˚-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. © 2025, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.
KW - approximative compactness
KW - approximative precompactness
KW - best approximation
KW - Chebyshev sun
KW - connected set
KW - Efimov–Stechkin space
KW - metrical exterior of a set
KW - projection boundary
KW - projection closed set
KW - sun
U2 - 10.21538/0134-4889-2025-31-3-20-35
DO - 10.21538/0134-4889-2025-31-3-20-35
M3 - статья
VL - 31
SP - 20
EP - 35
JO - ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УРО РАН
JF - ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УРО РАН
SN - 0134-4889
IS - 3
ER -
ID: 151303498