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

This article is cited in 20 scientific papers (total in 20 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).


DOI: https://doi.org/10.4213/rm9698

Full text: PDF file (1357 kB)
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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

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    Citing articles on Google Scholar: Russian citations, English citations
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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  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  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  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
    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  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  isi  elib
    17. Alimov A.R., “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. Alimov A.R., “Solarity of Sets in Max-Approximation Problems”, J. Fixed Point Theory Appl., 21:3 (2019), UNSP 76  crossref  isi
    19. Tsar'kov I.G., “Metric Max-Distance Function in Max-Approximation Problems”, Russ. J. Math. Phys., 26:2 (2019), 219–226  crossref  isi
    20. Alimov A.R. Shchepin E.V., “Convexity of Suns in Tangent Directions”, Dokl. Math., 99:1 (2019), 14–15  crossref  isi
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