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Section
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Mathematics
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Title
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$\alpha$-sets in finite dimensional Euclidean spaces and their properties
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Author(-s)
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Ushakov V.N.a,
Uspenskii A.A.a
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Affiliations
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Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa
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Abstract
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The concept of $\alpha$-set in a finite-dimensional Euclidean space, which is one of generalizations of the notion of a convex set, is introduced. The emergence of this concept is connected with the study of properties of attainability sets of nonlinear controlled systems. The numerical characteristic of nonconvexity degree of a set on the basis of which a classification of sets is carried out is defined in the paper. Analogs of basic concepts from the convex analysis are introduced into consideration and their properties are studied. Statements in the spirit of such theorems from the convex analysis as the theorem of existence of basic hyperplane to a convex set and theorems of separability of convex sets in Euclidean space are formulated and proved. The concept of magoriums of nonconvex sets is studied. Property of a magoriums is a sufficient condition for representation of a closed nonconvex set in the form of crossing of half-spaces in the sense of definitions entered in this work. The obtained results of the theory of separability of nonconvex sets can be extended on a case of hypograph and epigraph of the scalar functions with Lipschitz condition.
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Keywords
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convex set, convex hull, $\alpha$-set, $\alpha$-hyperplane, $\alpha$-separability
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UDC
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514.74
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MSC
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52A30
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DOI
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10.20537/vm160109
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Received
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21 December 2015
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Language
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Russian
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Citation
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Ushakov V.N., Uspenskii A.A. $\alpha$-sets in finite dimensional Euclidean spaces and their properties, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 95-120.
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References
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- Ushakov V.N., Uspenskii A.A., Fomin A.N. $\alpha$-mnozhestva i ikh svoistva ($\alpha$-sets and their properties), Ekaterinburg: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2004, 62 p.
- Bruce J.W., Giblin P.J. Krivye i osobennosti (Curves and singularities), Moscow: Mir, 1988, 262 p.
- Motzkin T. Sur quelques proprietes caracteristiques des ensembles convexes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 1935, vol. 21, pp. 562-567.
- Polovinkin E.S., Balashov M.B. Elementy vypuklogo i sil'no vypuklogo analiza (Elements of convex and strongly convex analysis), Moscow: Fizmatlit, 2007, 360 p.
- Pshenichnyi B.N. Vypuklyi analiz i ekstremal'nye zadachi (Convex analysis and extremal problems), Moscow: Nauka, 1980, 320 p.
- Dem'yanov V.F., Rubinov A.M. Osnovy negladkogo analiza i kvazidifferentsial'noe ischislenie (Fundamentals of nonsmooth analysis and quasidifferential calculus), Moscow: Nauka, 1990, 431 p.
- Bouligand G. Sur les surfaces depourvues de points hyperlimites, Ann. Soc. Polon. Math., 1930, vol. 9, pp. 32-41.
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