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Uspekhi Mat. Nauk, 2016, Volume 71, Issue 1(427), Pages 3–84 (Mi umn9698)  

This article is cited in 40 scientific papers (total in 40 papers)

Connectedness and solarity in problems of best and near-best approximation

A. R. Alimov, I. G. Tsar'kov

Faculty of Mechanics and Mathematics, Moscow State University

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.

Keywords: sun, strict sun, Chebyshev set, near-best approximation, connectedness, infinite connectedness, monotone path-connectedness, eikonal equation.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00295
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00295).


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English version:
Russian Mathematical Surveys, 2016, 71:1, 1–77

Bibliographic databases:

UDC: 517.982.256
MSC: Primary 41A65; Secondary 52A30, 54C60, 54C65, 78A05
Received: 02.10.2015

Citation: A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Uspekhi Mat. Nauk, 71:1(427) (2016), 3–84; Russian Math. Surveys, 71:1 (2016), 1–77

Citation in format AMSBIB
\by A.~R.~Alimov, I.~G.~Tsar'kov
\paper Connectedness and solarity in problems of best and near-best approximation
\jour Uspekhi Mat. Nauk
\yr 2016
\vol 71
\issue 1(427)
\pages 3--84
\jour Russian Math. Surveys
\yr 2016
\vol 71
\issue 1
\pages 1--77

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    This publication is cited in the following articles:
    1. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. R. Alimov, “Selections of the metric projection operator and strict solarity of sets with continuous metric projection”, Sb. Math., 208:7 (2017), 915–928  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. I. G. Tsar'kov, “Continuous selection from the sets of best and near-best approximation”, Dokl. Math., 96:1 (2017), 362–364  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    4. I. G. Tsar'kov, “Singular sets of surfaces”, Russ. J. Math. Phys., 24:2 (2017), 263–271  crossref  mathscinet  zmath  isi  scopus
    5. I. G. Tsar'kov, “Properties of $C^1$-solutions to the eikonal equation”, Lobachevskii J. Math., 38:4 (2017), 763–766  crossref  mathscinet  zmath  isi  scopus
    6. A. R. Alimov, “A monotone path-connected set with outer radially lower continuous metric projection is a strict sun”, Siberian Math. J., 58:1 (2017), 11–15  mathnet  crossref  crossref  isi  elib  elib
    7. A. R. Alimov, “On approximative properties of locally Chebyshev sets”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44:1 (2018), 36–42  mathscinet  zmath  isi
    8. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. A. R. Alimov, E. V. Shchepin, “Convexity of Chebyshev sets with respect to tangent directions”, Russian Math. Surveys, 73:2 (2018), 366–368  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734  mathnet  crossref  crossref  mathscinet  isi  elib
    12. I. G. Tsarkov, “Ustoichivost otnositelnogo chebyshëvskogo proektora v poliedralnykh prostranstvakh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 235–245  mathnet  crossref  elib
    13. A. R. Alimov, “Ogranichennaya styagivaemost strogikh solnts v trëkhmernykh prostranstvakh”, Fundament. i prikl. matem., 22:1 (2018), 3–11  mathnet  mathscinet
    14. A. A. Vasileva, “Kriterii suschestvovaniya $1$-lipshitsevoi vyborki iz metricheskoi proektsii na mnozhestvo iz nepreryvnykh vyborok iz mnogoznachnogo otobrazheniya”, Fundament. i prikl. matem., 22:1 (2018), 99–110  mathnet
    15. A. R. Alimov, “Selections of the best and near-best approximation operators and solarity”, Proc. Steklov Inst. Math., 303 (2018), 10–17  mathnet  crossref  crossref  mathscinet  isi  elib
    16. I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238  mathnet  crossref  crossref  mathscinet  isi  elib
    17. 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  crossref  mathscinet  zmath  isi  scopus
    18. A. R. Alimov, “Solarity of sets in max-approximation problems”, J. Fixed Point Theory Appl., 21:3 (2019), UNSP 76  crossref  isi
    19. I. G. Tsar'kov, “Metric max-distance function in max-approximation problems”, Russ. J. Math. Phys., 26:2 (2019), 219–226  crossref  isi
    20. A. R. Alimov, E. V. Shchepin, “Convexity of suns in tangent directions”, Dokl. Math., 99:1 (2019), 14–15  crossref  isi
    21. I. G. Tsar'kov, “Smooth solutions of the eikonal equation and the behaviour of local minima of the distance function”, Izv. Math., 83:6 (2019), 1234–1258  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    22. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008  mathnet  crossref  crossref  mathscinet  isi  elib
    23. A. R. Alimov, I. G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Russian Math. Surveys, 74:5 (2019), 775–849  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    24. I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    25. A. R. Alimov, “Singularities of solutions of the eikonal equation”, Differ. Equ., 55:10 (2019), 1311–1316  crossref  isi
    26. A. R. Alimov, V E. Shchepin, “Convexity of suns in tangent directions”, J. Convex Anal., 26:4 (2019), 1071–1076  isi
    27. K. S. Shklyaev, “A connected compact locally Chebyshev set in a finite-dimensional space is a Chebyshev set”, Sb. Math., 211:3 (2020), 455–465  mathnet  crossref  crossref  mathscinet  isi  elib
    28. A. R. Alimov, “Vypuklost i monotonnaya lineinaya svyaznost mnozhestv s nepreryvnoi metricheskoi proektsiei v trekhmernykh prostranstvakh”, Tr. IMM UrO RAN, 26, no. 2, 2020, 28–46  mathnet  crossref  mathscinet  elib
    29. I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Sb. Math., 211:8 (2020), 1190–1211  mathnet  crossref  crossref  isi  elib
    30. I. G. Tsar'kov, “The Geometry of a Singular Set of Hypersurfaces and the Eikonal Equation”, Math. Notes, 108:3 (2020), 426–433  mathnet  crossref  crossref  isi  elib
    31. A. R. Alimov, “Characterization of Sets with Continuous Metric Projection in the Space $\ell^\infty_n$”, Math. Notes, 108:3 (2020), 309–317  mathnet  crossref  crossref  mathscinet  isi  elib
    32. P. D. Lebedev, A. A. Uspenskii, “Elementy analiticheskogo konstruktora reshenii v klasse zadach upravleniya po bystrodeistviyu s tselevym mnozhestvom s razryvnoi kriviznoi granitsy”, Vestnik rossiiskikh universitetov. Matematika, 25:132 (2020), 370–386  mathnet  crossref
    33. A. A. Uspenskii, P. D. Lebedev, “Svoistva nestatsionarnykh psevdovershin kraevogo mnozhestva pri razryve gladkosti krivizny ego granitsy v zadache Dirikhle dlya uravneniya tipa eikonala”, Sib. elektron. matem. izv., 17 (2020), 2028–2044  mathnet  crossref
    34. A. R. Alimov, “Geometric construction of Chebyshev sets and suns in three-dimensional spaces with cylindrical norm”, Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 75:5 (2020), 209–215  mathnet  crossref  mathscinet  zmath  isi  elib
    35. I. G. Tsar'kov, “Properties of monotone path-connected sets”, Izv. Math., 85:2 (2021), 306–331  mathnet  crossref  crossref  isi  elib
    36. I. G. Tsar'kov, “Properties of Monotone Connected Sets”, Math. Notes, 109:5 (2021), 819–827  mathnet  crossref  crossref  isi  elib
    37. A. R. Alimov, B. B. Bednov, “Monotone path-connectedness of Chebyshev sets in three-dimensional spaces”, Sb. Math., 212:5 (2021), 636–654  mathnet  crossref  crossref  isi  elib
    38. P. D. Lebedev, A. A. Uspenskii, “Ob analiticheskom postroenii reshenii v odnom klasse zadach upravleniya po bystrodeistviyu s nevypuklym tselevym mnozhestvom”, Tr. IMM UrO RAN, 27, no. 3, 2021, 128–140  mathnet  crossref  elib
    39. A. A. Uspenskii, P. D. Lebedev, “On singularity structure of minimax solution to Dirichlet problem for eikonal type equation with discontinuous curvature of boundary of boundary set”, Ufa Math. J., 13:3 (2021), 126–151  mathnet  crossref  isi
    40. A. R. Alimov, I. G. Tsar'kov, “Approximatively Compact Sets in Asymmetric Efimov–Stechkin Spaces and Convexity of Almost Suns”, Math. Notes, 110:6 (2021), 947–951  mathnet  crossref  crossref
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