Geodesic
Geometry and the asymptotics of wave forms
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 27-37.

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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 $\alpha$-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.
Mots-clés : wavefront
Keywords: convex hull, pseudovertex, measure of nonconvexity, bisector of a set. 
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P. D. Lebedev; A. A. Uspenskii. Geometry and the asymptotics of wave forms. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 27-37. http://geodesic.mathdoc.fr/item/IVM_2008_3_a2/

[1] Leikhtveis K., Vypuklye mnozhestva, Nauka, M., 1985, 335 pp.

[2] Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979, 399 pp. | MR

[3] Singer I., Best approximation in normed spaces by elements of linear subspaces, Springer-Verlag, Berlin, 1970, 415 pp.

[4] Rashevskii P. K., Kurs differentsialnoi geometrii, Editorial-URSS, M., 2003, 432 pp.

[5] Uspenskii A. A., Ushakov V. N., Fomin A. N., $\alpha$-mnozhestva i ikh svoistva, Dep. v VINITI 02.04.04, No 543-V2004, In-t matem. i mekh. UrO RAN, Ekaterinburg, 2004, 62 pp. | Zbl

[6] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR | Zbl

[7] Arnold V. I., Teoriya katastrof, Izd-vo MGU, M., 1983, 80 pp.

[8] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., “Teoriya bifurkatsii”, Itogi nauki i tekhn. Sovremen. probl. matem. Fundam. napravleniya, 5, VINITI, M., 1986, 5–218 | MR

[9] Arnold V. I., Geometricheskie metody v teorii obyknovennykh differentsialnykh uravnenii, Izhevskaya respublikanskaya tipografiya, Izhevsk, 2000, 400 pp.

[10] Brus Dzh., Dzhiblin P., Krivye i osobennosti, Mir, M., 1988, 262 pp. | MR

[11] Sedykh V. D., “On the topology of symmetry sets of smooth submanifolds in $\mathbb{R}^k$”, Singularity Theory and Its Applications, Advanced Studies in Pure Mathematics, 43, 2006, 401–419 | MR | Zbl

[12] Lebedev P. D., Uspenskii A. A., “K voprosu o geometrii volnovykh frontov”, Izv. In-ta matem. i informatiki, no. 3 (37), UdGU, Izhevsk, 2006, 79–80

[13] Demyanov V. F., Vasilev L. V., Nedifferentsiruemaya optimizatsiya, Nauka, M., 1981, 384 pp.

[14] Uspenskii A. A., Analiticheskie metody vychisleniya mery nevypuklosti ploskikh mnozhestv, Dep. v VINITI 07.02.07, No 104-V2007, In-t matem. i mekh. UrO RAN, Ekaterinburg, 2007, 21 pp. | Zbl

[15] Blagodatskikh V. I., Vvedenie v optimalnoe upravlenie, Vyssh. shkola, M., 2001, 238 pp.