RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

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

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)
References: PDF file   HTML file

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

Bibliographic databases:

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

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
\Bibitem{AliTsa16} \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 \mathnet{http://mi.mathnet.ru/umn9698} \crossref{https://doi.org/10.4213/rm9698} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3507463} \zmath{https://zbmath.org/?q=an:06599754} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2016RuMaS..71....1A} \elib{http://elibrary.ru/item.asp?id=25707790} \transl \jour Russian Math. Surveys \yr 2016 \vol 71 \issue 1 \pages 1--77 \crossref{https://doi.org/10.1070/RM9698} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000376511100001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84973574839} 

• http://mi.mathnet.ru/eng/umn9698
• https://doi.org/10.4213/rm9698
• http://mi.mathnet.ru/eng/umn/v71/i1/p3

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. I. G. Tsar'kov, “Continuous selection from the sets of best and near-best approximation”, Dokl. Math., 96:1 (2017), 362–364
4. I. G. Tsar'kov, “Singular sets of surfaces”, Russ. J. Math. Phys., 24:2 (2017), 263–271
5. I. G. Tsar'kov, “Properties of $C^1$-solutions to the eikonal equation”, Lobachevskii J. Math., 38:4 (2017), 763–766
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
7. A. R. Alimov, “On approximative properties of locally Chebyshev sets”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44:1 (2018), 36–42
8. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859
9. A. R. Alimov, E. V. Shchepin, “Convexity of Chebyshev sets with respect to tangent directions”, Russian Math. Surveys, 73:2 (2018), 366–368
10. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579
11. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734
12. I. G. Tsarkov, “Ustoichivost otnositelnogo chebyshëvskogo proektora v poliedralnykh prostranstvakh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 235–245
13. A. R. Alimov, “Ogranichennaya styagivaemost strogikh solnts v trëkhmernykh prostranstvakh”, Fundament. i prikl. matem., 22:1 (2018), 3–11
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
15. A. R. Alimov, “Selections of the best and near-best approximation operators and solarity”, Proc. Steklov Inst. Math., 303 (2018), 10–17
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
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
18. Alimov A.R., “Solarity of Sets in Max-Approximation Problems”, J. Fixed Point Theory Appl., 21:3 (2019), UNSP 76
19. Tsar'kov I.G., “Metric Max-Distance Function in Max-Approximation Problems”, Russ. J. Math. Phys., 26:2 (2019), 219–226
20. Alimov A.R. Shchepin E.V., “Convexity of Suns in Tangent Directions”, Dokl. Math., 99:1 (2019), 14–15
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
22. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008
23. A. R. Alimov, I. G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Russian Math. Surveys, 74:5 (2019), 775–849
24. I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347
25. Alimov A.R., “Singularities of Solutions of the Eikonal Equation”, Differ. Equ., 55:10 (2019), 1311–1316
26. Alimov A.R. Shchepin V E., “Convexity of Suns in Tangent Directions”, J. Convex Anal., 26:4 (2019), 1071–1076
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
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
29. I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Sb. Math., 211:8 (2020), 1190–1211
30. I. G. Tsar'kov, “The Geometry of a Singular Set of Hypersurfaces and the Eikonal Equation”, Math. Notes, 108:3 (2020), 426–433
31. A. R. Alimov, “Characterization of Sets with Continuous Metric Projection in the Space $\ell^\infty_n$”, Math. Notes, 108:3 (2020), 309–317
•  Number of views: This page: 770 Full text: 124 References: 96 First page: 74