Abstract:
Sets admitting a continuous selection of the operators of best and near-best approximation are studied. Michael's classical continuous selection theorem is extended to the case of a lower semicontinuous metric projection in finite-dimensional spaces (with no a priori convexity conditions on its values). Sufficient conditions on the metric projection implying the solarity of the corresponding set are put forward in finite-dimensional polyhedral spaces. Available results for suns V are employed to establish the existence of continuous selections of the relative (with respect to V) Chebyshev near-centre map and of the sets of relative (with respect to V) near-Chebyshev points in certain classical spaces.
Bibliography: 30 titles.
Keywords:
set-valued mapping, continuous selection, sun, monotone path-connected set, relative Chebyshev centre and point.
\Bibitem{Tsa20}
\by I.~G.~Tsar'kov
\paper Approximative properties of sets and continuous selections
\jour Sb. Math.
\yr 2020
\vol 211
\issue 8
\pages 1190--1211
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\crossref{https://doi.org/10.1070/SM9319}
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Linking options:
https://www.mathnet.ru/eng/sm9319
https://doi.org/10.1070/SM9319
https://www.mathnet.ru/eng/sm/v211/i8/p132
This publication is cited in the following 25 articles:
I. G. Tsarkov, “Svoistva diskretnogo ne bolee chem schetnogo ob'edineniya mnozhestv v nesimmetrichnykh prostranstvakh”, Matem. sb., 216:2 (2025), 128–144
I.G. Tsar'kov, “Convexity of δ-Suns and γ-Suns in Asymmetric Spaces”, Russ. J. Math. Phys., 31:2 (2024), 325
A. R. Alimov, “Strogie solntsa, sostavlennye iz ploskostei”, Tr. IMM UrO RAN, 30, no. 4, 2024, 27–36
A. R. Alimov, “Strict Suns Composed of Planes”, Proc. Steklov Inst. Math., 327:S1 (2024), S1
I. G. Tsar'kov, “Connectedness in asymmetric spaces”, J. Math. Anal. Appl., 527:1 (2023), 127381
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, “Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces”, Izv. Math., 87:4 (2023), 835–851
A. R. Alimov, “Universality theorems for asymmetric spaces”, Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 26:02 (2023), 2250017
I. G. Tsar'kov, “Reflexivity for spaces with extended norm”, Russ. J. Math. Phys., 30:3 (2023), 399
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
B. B. Bednov, “Finite-Dimensional Spaces where the Class of Chebyshev Sets Coincides with the Class of Closed and Monotone Path-Connected Sets”, Math. Notes, 111:4 (2022), 505–514
I. G. Tsar'kov, “Continuity of a Metric Function and Projection in Asymmetric Spaces”, Math. Notes, 111:4 (2022), 616–623
A. R. Alimov, I. G. Tsar'kov, “Ball-complete sets and solar properties of sets in asymmetric spaces”, Results Math., 77:2 (2022), 86, 15 pp.
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, “Uniformly and locally convex asymmetric spaces”, Sb. Math., 213:10 (2022), 1444–1469
I. G. Tsar'kov, “Approximative and structural properties of sets in asymmetric spaces”, Izv. Math., 86:6 (2022), 1240–1253
I. G. Tsarkov, “Uniformly and locally convex asymmetric spaces”, Russ. J. Math. Phys., 29:1 (2022), 141–148
I. G. Tsar'kov, “Density of the Points of Continuity of the Metric Function and Projection in Asymmetric Spaces”, Math. Notes, 112:6 (2022), 1017–1024
A. R. Alimov, I. G. Tsar’kov, “Suns, moons, and ˚B -complete sets in asymmetric spaces”, Set-Valued Var. Anal., 30:3 (2022), 1233–1245
A. R. Alimov, I. G. Tsar'kov, “Solarity and proximinality in generalized rational approximation in spaces C(Q) and Lp”, Russ. J. Math. Phys., 29:3 (2022), 291–305
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