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Mat. Sb., 2000, Volume 191, Number 10, Pages 13–38 (Mi msb513)  

This article is cited in 14 scientific papers (total in 14 papers)

Uniform estimate of a compact convex set by a ball in an arbitrary norm

S. I. Dudov, I. V. Zlatorunskaya

Saratov State University named after N. G. Chernyshevsky

Abstract: The problem of the best uniform approximation of a compact convex set by a ball with respect to an arbitrary norm in the Hausdorff metric corresponding to that norm is considered.
The question is reduced to a convex programming problem, which can be studied by means of convex analysis. Necessary and sufficient conditions for the solubility of this problem are obtained and several properties of its solution are described. It is proved, in particular, that the centre of at least one ball of best approximation lies in the compact set under consideration; in addition, conditions ensuring that the centres of all balls of best approximation lie in this compact set and a condition for unique solubility are obtained.

DOI: https://doi.org/10.4213/sm513

Full text: PDF file (354 kB)
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English version:
Sbornik: Mathematics, 2000, 191:10, 1433–1458

Bibliographic databases:

UDC: 517.982.256+519.853.3
MSC: Primary 52A27; Secondary 49J52, 90C25
Received: 26.07.1999

Citation: S. I. Dudov, I. V. Zlatorunskaya, “Uniform estimate of a compact convex set by a ball in an arbitrary norm”, Mat. Sb., 191:10 (2000), 13–38; Sb. Math., 191:10 (2000), 1433–1458

Citation in format AMSBIB
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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. E. N. Sosov, “Best Approximations of Convex Compact Sets by Balls in the Hausdorff Metric”, Math. Notes, 76:2 (2004), 209–218  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. S. I. Dudov, I. V. Zlatorunskaya, “Best uniform approximation of a convex compact set by a ball in an arbitrary norm”, Comput. Math. Math. Phys., 45:3 (2005), 399–411  mathnet  mathscinet  zmath  elib  elib
    3. A. S. Dudova, “On the stability of the solution of the best approximation of a convex compact set by a ball”, Russian Math. (Iz. VUZ), 50:7 (2006), 22–30  mathnet  mathscinet  elib
    4. S. I. Dudov, A. B. Konoplev, “Approximation of Continuous Set-Valued Maps by Constant Set-Valued Maps with Image Balls”, Math. Notes, 82:4 (2007), 469–473  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. S. I. Dudov, A. S. Dudova, “On the stability of inner and outer approximations of a convex compact set by a ball”, Comput. Math. Math. Phys., 47:10 (2007), 1589–1602  mathnet  crossref  mathscinet  elib  elib
    6. S. I. Dudov, “Relations between several problems of estimating convex compacta by balls”, Sb. Math., 198:1 (2007), 39–53  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. S. I. Dudov, E. A. Mescheryakova, “O priblizhennom reshenii zadachi ob asferichnosti vypuklogo kompakta”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 10:4 (2010), 13–17  mathnet  elib
    8. S. I. Dudov, E. A. Mescheryakova, “Kharakterizatsiya ustoichivosti resheniya zadachi ob asferichnosti vypuklogo kompakta”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:2 (2011), 20–26  mathnet  elib
    9. S. I. Dudov, E. A. Meshcheryakova, “Method for finding an approximate solution of the asphericity problem for a convex body”, Comput. Math. Math. Phys., 53:10 (2013), 1483–1493  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    10. S. I. Dudov, M. A. Osiptsev, “O podkhode k priblizhennomu resheniyu zadachi nailuchshego priblizheniya vypuklogo tela sharom fiksirovannogo radiusa”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:3 (2014), 267–272  mathnet
    11. S. I. Dudov, “Systematization of problems on ball estimates of a convex compactum”, Sb. Math., 206:9 (2015), 1260–1280  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. S. I. Dudov, M. A. Osiptsev, “Ob ustoichivosti po funktsionalu resheniya zadachi o nailuchshem priblizhenii vypuklogo tela sharom fiksirovannogo radiusa”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:3 (2015), 273–279  mathnet  crossref  elib
    13. S. I. Dudov, E. A. Meshcheryakova, “On asphericity of convex body”, Russian Math. (Iz. VUZ), 59:2 (2015), 36–47  mathnet  crossref
    14. S. I. Dudov, M. A. Osiptsev, “Stability of best approximation of a convex body by a ball of fixed radius”, Comput. Math. Math. Phys., 56:4 (2016), 525–540  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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