Alpha sets and their hulls: analytical relationships in the plane case

The class of closed sets of two-dimensional Euclidean space that are not Chebyshev sets in common case is considered. Sets are studied from the standpoint of two well-known definitions that generalize the classical definition of a convex set. Within the framework of these definitions, analytical relationships are established between the parameters characterizing non-convex sets. A formula for calculating the function that determines the degree of non-convexity of a closed set, and a formula for calculating the radius of the support ball are found. The areas of application of the studied structures in the theory of control of dynamic systems are indicated. An illustrative example is given in which a procedure for analytical calculating the Chebyshev layer of a non-convex set with discontinuous curvature of its boundary is proposed.