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Convex trigonometry with applications to sub-Finsler geometry
Sbornik. Mathematics, Tome 210 (2019) no. 8, pp. 1179-1205 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new convenient method for describing flat convex compact sets and their polar sets is proposed. It generalizes the classical trigonometric functions $\sin$ and $\cos$. It is apparent that this method can be very useful for an explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Finsler problems with two-dimensional control lying in an arbitrary convex set $\Omega$ is investigated. Namely, problems on the Heisenberg, Engel, and Cartan groups and also Grushin's and Martinet's cases are considered. Particular attention is paid to the case when $\Omega$ is a convex polygon. Bibliography: 13 titles.
Keywords: sub-Finsler geometry, polar set, trigonometric functions, convex analysis, physical pendulum equation. 
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L. V. Lokutsievskiy. Convex trigonometry with applications to sub-Finsler geometry. Sbornik. Mathematics, Tome 210 (2019) no. 8, pp. 1179-1205. http://geodesic.mathdoc.fr/item/SM_2019_210_8_a4/

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