On asphericity of convex bodies

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• 作者：S. I. Dudov (1)
  E. A. Meshcheryakova (1)
  
  1. Saratov State University ; ul. Astrakhanskaya 83 ; Saratov ; 410012 ; Russia
  
• 关键词：asphericity ; convex body ; subdifferential ; quasiconvexity ; uniform bound
• 刊名：Russian Mathematics (Iz VUZ)
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：59
• 期：2
• 页码：36-47
• 全文大小：564 KB
• 参考文献：1. Bonnesen, T., Fenchel, W. / Theory of Convex Bodies (BCS Associates, 1987; Fazis, Moscow, 2002).
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  5. Dem鈥檡anov, V. F., Rubinov, A. M. / Foundations of Nonsmooth Analysis and Quasidifferential Calculus (Nauka, Moscow, 1990) [in Russian].
  6. Clarke, F. / Optimization and Nonsmooth Analysis (John Wiley & Sons, Inc., New York, 1983; Nauka, Moscow, 1986).
  7. Magaril-Il鈥檡aev, G. G., Tikhomirov, V. M., / Convex Analysis and Its Applications (Editorial URSS, Moscow, 2000) [in Russian].
  8. Polovinkin, E. S., Balashov, M. V., / Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2004) [in Russian].
  9. Ivanov, G. E. / Weakly Convex Sets and Functions (Fizmatlit, Moscow, 2006) [in Russian].
  10. Makeev, V. V. 鈥淎sphericity of Shadows of a Convex Body,鈥?J. Math. Sci., New York 140, No. 4, 535鈥?41 (2007). CrossRef
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  14. Vincze, St. 鈥溍渂er den Minimalkreisring einer Eilinie,鈥?Acta Sci. Math. (Szeged) 11, No. 3, 133鈥?38 (1947).
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  16. Kriticos, N. 鈥溍渂er convexe Flachen und einschlissende Kugeln,鈥?Math. Ann. 96, 583鈥?86 (1927). CrossRef
  17. Barany, I. 鈥淥n the Minimal Ring Containing the Boundary of Convex Body,鈥?Acta Sci. Math. (Szeged) 52, No. 1鈥?, 93鈥?00 (1988).
  18. Zucco, A. 鈥淢inimal Shell of a Typical Convex Body,鈥?Proc. Amer. Math. Soc. 109, 797鈥?02 (1990). CrossRef
  19. Nikol鈥檚kii, M. S., Silin, D. B. 鈥淥n the Best Approximation of Convex Compact Set by Addial Elements,鈥?Trudy Mat. Inst. Steklov 211, 338鈥?54 (1995).
  20. Dudov, S. I. 鈥淎pproximation of the Boundary of a Convex Compact by a Spherical Layer,鈥?Izv. Saratovsk. Univ. 1, No. 2, 64鈥?5 (2001).
  21. Dudov, S. I., Zlatorunskaya, I. V. 鈥淎 Uniform Estimate for a Compact Convex Set by a Ball of Arbitrary Norm,鈥?Sb. Math. 191, No. 9鈥?0, 1433鈥?458 (2000). CrossRef
  22. Dudov, S. I., Zlatorunskaya, I. V. 鈥淏est Approximation of a Compact Set by a Ball in an Arbitrary Norm,鈥?Adv. Math. Res. 2, 81鈥?14 (Nova Sci. Publ., Hauppauge, NY, 2003).
  23. Dudov, S. I., Zlatorunskaya, I. V. 鈥淥n an Approximate Uniform Estimate for a Compact Convex Set by a Ball of Arbitrary Norm,鈥?Comput. Math. Math. Phys. 45, No. 3, 399鈥?11 (2005).
  24. Dudov, S. I. 鈥淩elations Between Several Problems of Estimating Convex Compacts by Balls,鈥?Sb. Math. 198, No. 1, 39鈥?3 (2007). CrossRef
  25. Dudov, S. I. 鈥淪ubdifferentiability and Superdifferentiability of Distance Functions,鈥?Comput. Math. Math. Phys. 37, No. 8, 906鈥?12 (1997).
  26. Dudov, S. I., Meshcheryakova, E. A. 鈥淥n an Approximate Solution of the Problem of Aspherical Convex Compact Set,鈥?Izv. Saratovsk. Univ. Ser. Matem. Mekhan. Informatika 10, No. 4, 13鈥?7 (2010).
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Russian Library of Science
  
• 出版者：Allerton Press, Inc. distributed exclusively by Springer Science+Business Media LLC
• ISSN：1934-810X

文摘

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.