For a class of closed nonconvex sets in two-dimensional Euclidean space, an approach to finding the value of the Chebyshev layer is proposed. It is based on two well-known concepts that generalize the definition of a convex set. A family of planar sets with a finite number of pseudo-vertices is considered. Three sets of pseudo-vertices are selected for analysis. The sets differ from each other in the order of smoothness of the pseudo-vertices included in them. Within the framework of each of the three cases considered (the case of a piecewise smooth boundary of a set, the case of a discontinuity in the curvature of the boundary of a set, and the classical case when the curvature of the boundary is continuous), a formula for the limit value of the radii of the support balls (by Efimov and Stechkin) is found. We consider balls with centers lying on a branch of the bisector (on a one-dimensional manifold of the set of non-uniqueness) corresponding to the associated pseudo-vertex. The obtained formulas allow one to analytically calculate the value of the Chebyshev layer for nonconvex sets, including sets with a boundary of variable smoothness. An illustrative example and its interpretation from the point of view of optimal control theory are given.
Keywords: alpha set, set hull, metric projection, nonconvexity measure, set bisector, support ball, Chebyshev layer, control.
Received June 4, 2025
Revised July 9, 2025
Accepted July 14, 2025
Funding Agency: The study of the first author was supported by the Russian Science Foundation (project no. 25-11-00269, https://rscf.ru/project/25-11-00269/).
Aleksandr Aleksandrovich Uspenskii, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: uspen@imm.uran.ru
Pavel Dmitrievich Lebedev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: pleb@yandex.ru
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Cite this article as: A.A. Uspenskii, P.D. Lebedev. Finding the value of the Chebyshev layer of a flat set using constructions of the theory of alpha sets and Efimov–Stechkin support balls. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 3, pp. 264–280.
