Abstract:
We study properties of sets for which there is a continuous selection from
the set of almost-best approximants. We establish relations between local
and global selections, give various examples of sets possessing
a continuous εε-selection, and introduce the notions
of moduli of approximative continuity, approximative δδ-solarity
and uniform approximative continuity. These notions enable us to establish
the δδ-solarity of sets under certain conditions.
\Bibitem{Tsa16}
\by I.~G.~Tsar'kov
\paper Local and global continuous $\varepsilon$-selection
\jour Izv. Math.
\yr 2016
\vol 80
\issue 2
\pages 442--461
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\crossref{https://doi.org/10.1070/IM8348}
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Linking options:
https://www.mathnet.ru/eng/im8348
https://doi.org/10.1070/IM8348
https://www.mathnet.ru/eng/im/v80/i2/p165
This publication is cited in the following 21 articles:
A. R. Alimov, K. S. Ryutin, I. G. Tsar'kov, “Existence, uniqueness, and stability of best and near-best approximations”, Russian Math. Surveys, 78:3 (2023), 399–442
I. G. Tsar'kov, “Solarity and connectedness of sets in the space $C[a,b]$ and in finite-dimensional polyhedral spaces”, Sb. Math., 213:2 (2022), 268–282
A. R. Alimov, I. G. Tsar'kov, “Some Classical Problems of Geometric Approximation Theory in Asymmetric Spaces”, Math. Notes, 112:1 (2022), 3–16
I. G. Tsar'kov, “Approximative and structural properties of sets in asymmetric spaces”, Izv. Math., 86:6 (2022), 1240–1253
I. G. Tsar'kov, “Properties of monotone path-connected sets”, Izv. Math., 85:2 (2021), 306–331
I. G. Tsar'kov, “Properties of Suns in the Spaces l-1 and C(Q)”, Russ. J. Math. Phys., 28:3 (2021), 398–405
I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Sb. Math., 211:8 (2020), 1190–1211
I. G. Tsar’kov, “Some Applications of Geometric Theory of Approximations”, J Math Sci, 245:1 (2020), 64
A. R. Alimov, “Continuity of the metric projection and local solar properties of sets: continuity of the metric projection and solar properties”, Set-Valued Var. Anal., 27:1 (2019), 213–222
I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008
I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347
F. F. Komarov, O. V. Milchanin, I. N. Parkhomenko, L. A. Vlasukova, N. S. Nechaev, V. A. Skuratov, V. N. Yuvchenko, “Influence of the Irradiation with High-Energy Xe Ions on the Structure and Photoluminescence of Silicon and Silica with InAs Nanoclusters”, J Eng Phys Thermophy, 92:2 (2019), 508
I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859
I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579
I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734
A. R. Alimov, “Selections of the best and near-best approximation operators and solarity”, Proc. Steklov Inst. Math., 303 (2018), 10–17
I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238
A. R. Alimov, “Bounded contractibility of strict suns in three-dimensional spaces”, J. Math. Sci., 250:3 (2020), 385–390
I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669
I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049
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