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On asphericity of convex bodies

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Abstract

The paper deals with a finite-dimensional problem of minimizing the ratio of the radius of the sphere circumscribed about a given convex body (in an arbitrary norm) to the radius of the inscribed sphere. The minimization is performed by choosing a common center of these spheres. We prove that the objective function of this problem is quasiconvex and subdifferentiable and establish a criterion for the unique solvability of the problem. The considered problem is compared with those close to it in geometric sense.

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References

  1. Bonnesen, T., Fenchel, W. Theory of Convex Bodies (BCS Associates, 1987; Fazis, Moscow, 2002).

    MATH  Google Scholar 

  2. Rockafellar, R. T. Convex Analysis (Princeton University Press, 1970;Mir, Moscow, 1973).

    MATH  Google Scholar 

  3. Pshenichnyi, B. N, Convex Analysis and Extremal Problems (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  4. Dem’yanov, V. F., Vasil’ev, L. V. Nondifferential Optimization (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

  5. Dem’yanov, V. F., Rubinov, A. M. Foundations of Nonsmooth Analysis and Quasidifferential Calculus (Nauka, Moscow, 1990) [in Russian].

    MATH  Google Scholar 

  6. Clarke, F. Optimization and Nonsmooth Analysis (John Wiley & Sons, Inc., New York, 1983; Nauka, Moscow, 1986).

    MATH  Google Scholar 

  7. Magaril-Il’yaev, G. G., Tikhomirov, V. M., Convex Analysis and Its Applications (Editorial URSS, Moscow, 2000) [in Russian].

    Google Scholar 

  8. Polovinkin, E. S., Balashov, M. V., Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2004) [in Russian].

    Google Scholar 

  9. Ivanov, G. E. Weakly Convex Sets and Functions (Fizmatlit, Moscow, 2006) [in Russian].

    MATH  Google Scholar 

  10. Makeev, V. V. “Asphericity of Shadows of a Convex Body,” J. Math. Sci., New York 140, No. 4, 535–541 (2007).

    Article  MathSciNet  Google Scholar 

  11. Kamenev, G. G. “The Initial Convergence Rate of Adaptive Methods for Polyhedral Approximation of Convex Bodies,” Comput. Math. Math. Phys. 48, No. 5, 724–738 (2008).

    Article  MathSciNet  Google Scholar 

  12. D’Ocagne, M. “Sur Certaine Figures Minimales,” Bull. Soc. Math. France 12, 168–177 (1884).

    MATH  MathSciNet  Google Scholar 

  13. Lebesque, H. “Sur Quelques Questions de Minimum, Relatives and Courbes Orbiformes, et sur Leurs Rapports Avec le Calcul des Cariations,” J. Math. 4, No. 8, 67–96 (1921).

    Google Scholar 

  14. Vincze, St. “Über den Minimalkreisring einer Eilinie,” Acta Sci. Math. (Szeged) 11, No. 3, 133–138 (1947).

    MATH  MathSciNet  Google Scholar 

  15. Vincze, I. “Über Kreisringe, die eine Eilinie einschlissen,” Studia Sci. Math. Hungar. 9, No. 1–2, 155–159 (1974).

    MathSciNet  Google Scholar 

  16. Kriticos, N. “Über convexe Flachen und einschlissende Kugeln,” Math. Ann. 96, 583–586 (1927).

    Article  MathSciNet  Google Scholar 

  17. Barany, I. “On the Minimal Ring Containing the Boundary of Convex Body,” Acta Sci. Math. (Szeged) 52, No. 1–2, 93–100 (1988).

    MATH  MathSciNet  Google Scholar 

  18. Zucco, A. “Minimal Shell of a Typical Convex Body,” Proc. Amer. Math. Soc. 109, 797–802 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  19. Nikol’skii, M. S., Silin, D. B. “On the Best Approximation of Convex Compact Set by Addial Elements,” Trudy Mat. Inst. Steklov 211, 338–354 (1995).

    MathSciNet  Google Scholar 

  20. Dudov, S. I. “Approximation of the Boundary of a Convex Compact by a Spherical Layer,” Izv. Saratovsk. Univ. 1, No. 2, 64–75 (2001).

    Google Scholar 

  21. Dudov, S. I., Zlatorunskaya, I. V. “A Uniform Estimate for a Compact Convex Set by a Ball of Arbitrary Norm,” Sb. Math. 191, No. 9–10, 1433–1458 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  22. Dudov, S. I., Zlatorunskaya, I. V. “Best Approximation of a Compact Set by a Ball in an Arbitrary Norm,” Adv. Math. Res. 2, 81–114 (Nova Sci. Publ., Hauppauge, NY, 2003).

    MathSciNet  Google Scholar 

  23. Dudov, S. I., Zlatorunskaya, I. V. “On an Approximate Uniform Estimate for a Compact Convex Set by a Ball of Arbitrary Norm,” Comput. Math. Math. Phys. 45, No. 3, 399–411 (2005).

    MathSciNet  Google Scholar 

  24. Dudov, S. I. “Relations Between Several Problems of Estimating Convex Compacts by Balls,” Sb. Math. 198, No. 1, 39–53 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  25. Dudov, S. I. “Subdifferentiability and Superdifferentiability of Distance Functions,” Comput. Math. Math. Phys. 37, No. 8, 906–912 (1997).

    MathSciNet  Google Scholar 

  26. Dudov, S. I., Meshcheryakova, E. A. “On an Approximate Solution of the Problem of Aspherical Convex Compact Set,” Izv. Saratovsk. Univ. Ser. Matem. Mekhan. Informatika 10, No. 4, 13–17 (2010).

    Google Scholar 

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Authors and Affiliations

  1. Saratov State University, ul. Astrakhanskaya 83, Saratov, 410012, Russia

    S. I. Dudov & E. A. Meshcheryakova

Authors
  1. S. I. Dudov
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  2. E. A. Meshcheryakova
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Correspondence to S. I. Dudov.

Additional information

Original Russian Text © S.I. Dudov, E.A. Meshcheryakova, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 2, pp. 45–58.

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Dudov, S.I., Meshcheryakova, E.A. On asphericity of convex bodies. Russ Math. 59, 36–47 (2015). https://doi.org/10.3103/S1066369X15020061

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  • DOI: https://doi.org/10.3103/S1066369X15020061

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