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Russia Yekaterinburg
Section  Mathematics 
Title  $\alpha$sets in finite dimensional Euclidean spaces and their properties 
Author(s)  Ushakov V.N.^{a}, Uspenskii A.A.^{a} 
Affiliations  Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences^{a} 
Abstract  The concept of $\alpha$set in a finitedimensional 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 halfspaces 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. 
Keywords  convex set, convex hull, $\alpha$set, $\alpha$hyperplane, $\alpha$separability 
UDC  514.74 
MSC  52A30 
DOI  10.20537/vm160109 
Received  21 December 2015 
Language  Russian 
Citation  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. 95120. 
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