Abstract
Methods of convex analysis and differential geometry are applied to the study of properties of nonconvex sets in the plane. Constructions of the theory of α-sets are used as a tool for investigation of problems of the control theory and the theory of differential games. The notions of the bisector and of a pseudovertex of a set introduced in the paper, which allow ones to study the geometry of sets and compute their measure of nonconvexity, are of independent interest. These notions are also useful in studies of evolution of sets of attainability of controllable systems and in constructing of wavefronts. In this paper, we develop a numerically-analytical approach to finding pseudovertices of a curve, computation of the measure of nonconvexity of a plane set, and constructing front sets on the basis these data.
In the paper, we give the results of numerical constructing of bisectors and wavefronts for plane sets. We demonstrate the relation between nonsmoothness of wavefronts and singularity of the geometry of the initial set. We also single out a class of sets whose bisectors have an asymptote.
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Original Russian Text © P.D. Lebedev, A.A. Uspenskii, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 3, pp. 27–37.
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Lebedev, P.D., Uspenskii, A.A. Geometry and asymptotics of wavefronts. Russ Math. 52, 24–33 (2008). https://doi.org/10.3103/S1066369X08030031
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DOI: https://doi.org/10.3103/S1066369X08030031