Generalized Convex Envelopes of Sets and the Problem of Shadow
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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.
KeywordsEuclidean space sphere ball convexity linear convexity
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- 2.L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis [in Russian], Nauka, Novosibirsk, 1979.Google Scholar
- 4.M. Andersson, M. Passare, and R. Sigurdsson, Complex Convexity and Analytic Functionals, Basel, Birkhäuser, 2005.Google Scholar
- 6.L. Hörmander, Notions of Convexity, Birkhäuser, Basel, 2007.Google Scholar
- 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
- 10.V. P. Soltan, Introduction into the Axiomatic Theory of Convexity [in Russian], Shtiintsa, Kishinev, 1984.Google Scholar
- 11.Yu. B. Zelinskii, Multivalued Mappings in Analysis [in Russian], Naukova Dumka, Kiev, 1993.Google Scholar
- 12.Yu. B. Zelinskii, Convexity. Selected Chapters [in Russian], Inst. of Mathem. of the NASU, Kiev, 2012.Google Scholar
- 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.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.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