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Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 5, Pages 19–46 (Mi izv4280)  

This article is cited in 7 scientific papers (total in 7 papers)

On the convexity of $N$-Chebyshev sets

P. A. Borodin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We define $N$-Chebyshev sets in a Banach space $X$ for every positive integer $N$ (when $N=1$, these are ordinary Chebyshev sets) and study conditions that guarantee their convexity. In particular, we prove that all $N$-Chebyshev sets are convex when $N$ is even and $X$ is uniformly convex or $N\ge 3$ is odd and $X$ is smooth uniformly convex.

Keywords: Chebyshev set, convexity problem.

DOI: https://doi.org/10.4213/im4280

Full text: PDF file (647 kB)
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English version:
Izvestiya: Mathematics, 2011, 75:5, 889–914

Bibliographic databases:

UDC: 517.982.256
MSC: 46B20, 41A50, 41A65
Received: 29.12.2009
Revised: 03.06.2010

Citation: P. A. Borodin, “On the convexity of $N$-Chebyshev sets”, Izv. RAN. Ser. Mat., 75:5 (2011), 19–46; Izv. Math., 75:5 (2011), 889–914

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    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. P. A. Borodin, “Density of a semigroup in a Banach space”, Izv. Math., 78:6 (2014), 1079–1104  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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
    3. 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
    4. Namboodiri M.N.N., Pramod S., Vijayarajan A.K., “Cebysev Subspaces of $C^*$-Algebras - a Survey”, Operator Algebras and Mathematical Physics, Operator Theory Advances and Applications, 247, eds. Bhattacharyya T., Dritschel M., Birkhauser Verlag Ag, 2015, 101–121  crossref  mathscinet  zmath  isi
    5. B. B. Bednov, “The $n$-antiproximinal sets”, Moscow University Mathematics Bulletin, 70:3 (2015), 130–135  mathnet  crossref  mathscinet
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. B. B. Bednov, “Example of an antiproximinal, but not a 2-antiproximinal convex closed bounded body”, Moscow University Mathematics Bulletin, 71:5 (2016), 208–211  mathnet  crossref  mathscinet  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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