Abstract:
This survey is concerned with structural characteristics of ‘suns’ in normed linear spaces, with special emphasis on connectedness and monotone path-connectedness. Consideration is given to both direct theorems in geometric approximation theory in which approximative properties of sets are derived from their structural characteristics, and converse theorems in which structural properties of sets are derived from their approximative characteristics. Geometric methods of approximation theory are employed in solving the eikonal equation.
Bibliography: 231 titles.
Citation:
A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77
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\by A.~R.~Alimov, I.~G.~Tsar'kov
\paper Connectedness and solarity in problems of best and near-best approximation
\jour Russian Math. Surveys
\yr 2016
\vol 71
\issue 1
\pages 1--77
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This publication is cited in the following 87 articles:
A. R. Alimov, “Hereditary properties of lower semicontinuous metric projection and solar properties of sets”, J Anal, 2025
I. G. Tsar'kov, “θ-metric function
in the problem of minimization of functionals”, Izv. Math., 88:2 (2024), 369–388
P. D. Lebedev, A. A. Uspenskii, “Metod Nyutona pri postroenii singulyarnogo mnozhestva minimaksnogo resheniya v odnom klasse kraevykh zadach dlya uravnenii Gamiltona — Yakobi”, Chelyab. fiz.-matem. zhurn., 9:1 (2024), 63–76
K. S. Shklyaev, “On Locally Chebyshev Sets”, Math. Notes, 115:4 (2024), 636–641
A. R. Alimov, I. G. Tsar'kov, “Chebyshev unions of planes, and their approximative and geometric properties”, J. Approx. Theory, 298 (2024), 106009–12
A. A. Uspenskii, P. D. Lebedev, “Alfa-mnozhestva i ikh obolochki:analiticheskie vzaimosvyazi v ploskom sluchae”, Vestnik rossiiskikh universitetov. Matematika, 29:146 (2024), 204–217
I.G. Tsar'kov, “Convexity of δ-Suns and γ-Suns in Asymmetric Spaces”, Russ. J. Math. Phys., 31:2 (2024), 325
P. A. Borodin, E. A. Savinova, “Any Chebyshev curve without self-intersections is monotone”, Math. Notes, 116:2 (2024), 387–389
P. D. Lebedev, A. A. Uspenskii, “Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case”, Sb. Math., 215:9 (2024), 1224–1248
I.G. Tsarkov, “Relations Between Various Types of Suns in Asymmetric Spaces”, Russ. J. Math. Phys., 31:3 (2024), 562
A. R. Alimov, I. G. Tsar'kov, “Chebyshev sets composed of subspaces in asymmetric normed spaces”, Izv. Math., 88:6 (2024), 1032–1049
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
E. A. Savinova, “Sets in Rn monotone path-connected with respect to some norm”, Moscow University Mathematics Bulletin, 78:1 (2023), 49–51
Alexey R. Alimov, “Strict Protosuns in Asymmetric Spaces of Continuous Functions”, Results Math, 78:3 (2023)
I. G. Tsar’kov, “Estimates of the Chebyshev Radius in Terms of the MAX-Metric Function and the MAX-Projection Operator”, Russ. J. Math. Phys., 30:1 (2023), 128
I.G. Tsar'kov, “Connectedness in asymmetric spaces”, Journal of Mathematical Analysis and Applications, 527:1 (2023), 127381
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