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Uspekhi Mat. Nauk, 1996, Volume 51, Issue 6(312), Pages 125–188 (Mi umn1020)  

This article is cited in 52 scientific papers (total in 53 papers)

The problem of convexity of Chebyshev sets

V. S. Balaganskii, L. P. Vlasov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

DOI: https://doi.org/10.4213/rm1020

Full text: PDF file (726 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 1996, 51:6, 1127–1190

Bibliographic databases:

UDC: 517.98+517.518.826
MSC: 41A50, 58C20, 46B28, 47H05
Received: 12.09.1996

Citation: V. S. Balaganskii, L. P. Vlasov, “The problem of convexity of Chebyshev sets”, Uspekhi Mat. Nauk, 51:6(312) (1996), 125–188; Russian Math. Surveys, 51:6 (1996), 1127–1190

Citation in format AMSBIB
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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. A. R. Alimov, “Vsyakoe li chebyshevskoe mnozhestvo vypuklo?”, Matem. prosv., ser. 3, 2, MTsNMO, M., 1998, 155–172  mathnet
    2. U. Westphal, T. Schwartz, “Farthest points and monotone operators”, BAZ, 58:1 (1998), 75  crossref  mathscinet  zmath  isi
    3. A. R. Alimov, M. I. Karlov, “Sets with External Chebyshev Layer”, Math. Notes, 69:2 (2001), 269–273  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. R. Alimov, “On the Structure of the Complements of Chebyshev Sets”, Funct. Anal. Appl., 35:3 (2001), 176–182  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. S. Balaganskii, “Necessary Conditions for Differentiability of Distance Functions”, Math. Notes, 72:6 (2002), 752–756  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Wu, ZL, “A Chebyshev set and its distance function”, Journal of Approximation Theory, 119:2 (2002), 181  crossref  mathscinet  isi  elib  scopus  scopus
    7. Alimov, AR, “Characterisations of Chebyshev sets in c(0)”, Journal of Approximation Theory, 129:2 (2004), 217  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. A. R. Alimov, “Convexity of Chebyshev Sets Contained in a Subspace”, Math. Notes, 78:1 (2005), 3–13  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. A. R. Alimov, “The Geometric Structure of Chebyshev Sets in $\ell^\infty(n)$”, Funct. Anal. Appl., 39:1 (2005), 1–8  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. A. R. Alimov, “Connectedness of suns in the space $c_0$”, Izv. Math., 69:4 (2005), 651–666  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. Cobzas, S, “Geometric properties of Banach spaces and the existence of nearest and farthest points”, Abstract and Applied Analysis, 2005, no. 3, 259  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. A. R. Alimov, “Monotone path-connectedness of Chebyshev sets in the space $C(Q)$”, Sb. Math., 197:9 (2006), 1259–1272  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. V. A. Koshcheev, “Vlasov–Wulbert connectedness of sets in Hilbert space”, Math. Notes, 80:5 (2006), 744–747  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    14. Duda J., “On the Size of the Set of Points Where the Metric Projection Is Discontinuous”, J. Nonlinear Convex Anal., 7:1 (2006), 67–70  mathscinet  zmath  isi
    15. I. A. Pyatyshev, “Operations on Approximatively Compact Sets”, Math. Notes, 82:5 (2007), 653–659  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. I. A. Pyatyshev, “An example of a bounded approximatively compact set which is not locally compact”, Russian Math. Surveys, 62:5 (2007), 1007–1008  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. Hiriart-Urruty, JB, “Potpourri of conjectures and open questions in nonlinear analysis and optimization”, SIAM Review, 49:2 (2007), 255  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    18. Francesca Faraci, Antonio Iannizzotto, “Well Posed Optimization Problems and Nonconvex Chebyshev Sets in Hilbert Spaces”, SIAM J Optim, 19:1 (2008), 211  crossref  mathscinet  zmath  isi  scopus  scopus
    19. T.D. Narang, Shavetambry Tejpal, “On δ-suns”, Publ. Inst. Math. (Belgr.), 83:97 (2008), 99  crossref  mathscinet  zmath  scopus  scopus
    20. P. A. Borodin, I. A. Pyatyshev, “An Example of Non-Approximatively-Compact Existence Set with Finite-Valued Metric Projection”, Math. Notes, 86:2 (2009), 159–162  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. M. V. Balashov, G. E. Ivanov, “Weakly convex and proximally smooth sets in Banach spaces”, Izv. Math., 73:3 (2009), 455–499  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    22. G. E. Ivanov, “Farthest Points and Strong Convexity of Sets”, Math. Notes, 87:3 (2010), 355–366  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    23. J. P. Revalski, N. V. Zhivkov, “Small sets in best approximation theory”, J Global Optim, 2010  crossref  mathscinet  zmath  isi  scopus  scopus
    24. T.D. Narang, R. Sangeeta, “A note on suns in convex metric spaces”, Publ. Inst. Math. (Belgr.), 87:101 (2010), 139  crossref  mathscinet  zmath  isi  scopus  scopus
    25. Li Ch., Song W., Yao J.-Ch., “The Bregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces”, Journal of Approximation Theory, 162:6 (2010), 1128–1149  crossref  mathscinet  zmath  isi  scopus  scopus
    26. I. G. Tsar'kov, “Properties of Sets Admitting Stable $\varepsilon$-Selections”, Math. Notes, 89:4 (2011), 572–576  mathnet  crossref  crossref  mathscinet  isi
    27. Chao Kan, Wen Song, “The Moreau envelope function and proximal mapping in the sense of the Bregman distance”, Nonlinear Analysis: Theory, Methods & Applications, 2011  crossref  mathscinet  isi  scopus  scopus
    28. P. A. Borodin, “On the convexity of $N$-Chebyshev sets”, Izv. Math., 75:5 (2011), 889–914  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    29. P. A. Borodin, “$2$-Chebyshev Subspaces in the Spaces $L_1$ and $C$”, Math. Notes, 91:6 (2012), 770–781  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    30. Abderrahim Jourani, Lionel Thibault, Dariusz Zagrodny, “Differential properties of the Moreau envelope”, Journal of Functional Analysis, 2013  crossref  mathscinet  isi  scopus  scopus
    31. Nashine H.K. Kumam P. Vetro C., “Best Proximity Point Theorems for Rational Proximal Contractions”, Fixed Point Theory Appl., 2013, 95  crossref  mathscinet  zmath  isi  scopus  scopus
    32. JAMES FLETCHER, W.B.. MOORS, “Chebyshev sets”, J. Aust. Math. Soc, 2014, 1  crossref  mathscinet
    33. A. R. Alimov, “Vypuklost ogranichennykh chebyshevskikh mnozhestv v konechnomernykh prostranstvakh s nesimmetrichnoi normoi”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(2) (2014), 489–497  mathnet  crossref  elib
    34. Kutbi M.A., Chandok S., Sintunavarat W., “Optimal Solutions For Nonlinear Proximal C-N-Contraction Mapping in Metric Space”, J. Inequal. Appl., 2014, 193  crossref  mathscinet  zmath  isi  scopus  scopus
    35. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    36. Zagrodny D., “on Closures of Preimages of Metric Projection Mappings in Hilbert Spaces”, 23, no. 4, SI, 2015, 581–612  crossref  mathscinet  zmath  isi  scopus  scopus
    37. Garcia-Pacheco F.J., Hill J.R., “a Partial Positive Solution To a Conjecture of Ricceri”, 46, no. 1, 2015, 57–67  mathscinet  zmath  isi
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    39. de Lima R.F., “Geometrical and Analytical Properties of Chebyshev Sets in Riemannian Manifolds”, 144, no. 4, 2016, 1697–1710  crossref  mathscinet  zmath  isi  scopus  scopus
    40. Luo Zh., Sun L., Zhang W., “a Remark on the Stability of Approximative Compactness”, 2016, 2734947  crossref  mathscinet  zmath  isi  scopus  scopus
    41. Domokos A., Ingram J.M., Marsh M.M., “Projections Onto Closed Convex Sets in Hilbert Spaces”, Acta Math. Hung., 152:1 (2017), 114–129  crossref  mathscinet  zmath  isi  scopus  scopus
    42. Tsar'kov I.G., “Properties of C-1-Solutions to the Eikonal Equation”, Lobachevskii J. Math., 38:4, SI (2017), 763–766  crossref  mathscinet  zmath  isi  scopus  scopus
    43. Dutta S., Shunmugaraj P., “Weakly Compactly Lur Banach Spaces”, J. Math. Anal. Appl., 458:2 (2018), 1203–1213  crossref  mathscinet  zmath  isi  scopus  scopus
    44. Moors W.B., “Nearly Chebyshev Sets Are Almost Convex”, Set-Valued Var. Anal., 26:1 (2018), 67–76  crossref  mathscinet  zmath  isi  scopus  scopus
    45. Zhang Ya., Luo X.-F., Meng L., Wen Ch.-F., “Criteria For Convexity of Generalized Chebyshev Sets in Banach Spaces”, J. Nonlinear Convex Anal., 19:1, SI (2018), 167–177  mathscinet  zmath  isi
    46. 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
    47. 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
    48. Luo X.-F., Li Ch., Peng L., Yao J.-Ch., “Convexity of Generalized Proximinal Sets in Banach Spaces”, Optimization, 67:9, SI (2018), 1535–1551  crossref  mathscinet  zmath  isi  scopus
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    53. 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
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