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Journal of Mathematical Sciences

, Volume 211, Issue 5, pp 710–717 | Cite as

Generalized Convex Envelopes of Sets and the Problem of Shadow

  • Yurii B. Zelinskii
Article

Abstract

The principal goal of the present work is to solve the problem of shadow for any convex set with nonempty interior in the n-dimensional Euclidean space and under the action of a group of transformations. This problem can be considered as the determination of conditions ensuring the membership of a point to a generalized convex envelope of the family of sets obtained from the initial set by the action of the group of transformations.

Keywords

Euclidean space sphere ball convexity linear convexity 

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References

  1. 1.
    L. A. Aizenberg, “On the expansion of holomorphic functions of many complex variables into simple fractions,” Sibir. Mat. Zh., 8, No. 5, 1124–1142 (1967).MATHMathSciNetGoogle Scholar
  2. 2.
    L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis [in Russian], Nauka, Novosibirsk, 1979.Google Scholar
  3. 3.
    R. D. Anderson and V. L. Klee, “Convex functions and upper-semi-continuous collections,” Duke Math. J., 19, 349–357 (1952).MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Andersson, M. Passare, and R. Sigurdsson, Complex Convexity and Analytic Functionals, Basel, Birkhäuser, 2005.Google Scholar
  5. 5.
    H. Behnke and E. Peschl, “Zur Theorie der Funktionen mehrerer komplexer Veränderlichen Konvexität in bezug auf analytische Ebenen im kleinen und großen,” Math. Ann., 111, No. 2, 158–177 (1935).MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Hörmander, Notions of Convexity, Birkhäuser, Basel, 2007.Google Scholar
  7. 7.
    G. Khudaiberganov, On the Homogeneous Polynomially Convex Envelope of a Union of Balls [in Russian], Manuscr. dep. 21.02.1982, No. 1772, 85 Dep., VINITI, Moscow, 1982.Google Scholar
  8. 8.
    K. Leichtweiss, Konvexe Mengen, Springer, Berlin, 1980.CrossRefGoogle Scholar
  9. 9.
    A. Martineau, “Sur la topologie des espaces de fonctions holomorphes,” Math. Ann., 163, No. 1, 62–88 (1966).MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    V. P. Soltan, Introduction into the Axiomatic Theory of Convexity [in Russian], Shtiintsa, Kishinev, 1984.Google Scholar
  11. 11.
    Yu. B. Zelinskii, Multivalued Mappings in Analysis [in Russian], Naukova Dumka, Kiev, 1993.Google Scholar
  12. 12.
    Yu. B. Zelinskii, Convexity. Selected Chapters [in Russian], Inst. of Mathem. of the NASU, Kiev, 2012.Google Scholar
  13. 13.
    Yu. B. Zelinskii, “The problem of shadow for a family of sets,” Zbirn. Prats Inst. Mat. NANU, 12, No. 4 (2015).Google Scholar
  14. 14.
    Yu. B. Zelinskii, “The problem of the shadows,” Bulletin de la societ´e des sci. et letters de L´od´z (in print).Google Scholar
  15. 15.
    Yu. B. Zelinskii, I. Yu. Vygovskaya, and M. V. Stefanchuk, “Generalized convex sets and the problem of shadow” [in Russian], arXiv preprint arXiv:1501.06747.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of Mathematics of the NAS of UkraineKievUkraine

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