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Mat. Sb., 2019, Volume 210, Number 8, Pages 120–148 (Mi msb9134)  

Convex trigonometry with applications to sub-Finsler geometry

L. V. Lokutsievskiyab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

Abstract: 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00805-а
17-01-00809-а
This research was carried out with the support of the Russian Foundation for Basic Research (grant nos. 17-01-00805-a and 17-01-00809-a).


DOI: https://doi.org/10.4213/sm9134

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English version:
Sbornik: Mathematics, 2019, 210:8, 1179–1205

UDC: 514.172+517.977+514.13
MSC: 26A99, 49J30, 53C17
Received: 17.05.2018 and 26.10.2018

Citation: L. V. Lokutsievskiy, “Convex trigonometry with applications to sub-Finsler geometry”, Mat. Sb., 210:8 (2019), 120–148; Sb. Math., 210:8 (2019), 1179–1205

Citation in format AMSBIB
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\by L.~V.~Lokutsievskiy
\paper Convex trigonometry with applications to sub-Finsler geometry
\jour Mat. Sb.
\yr 2019
\vol 210
\issue 8
\pages 120--148
\mathnet{http://mi.mathnet.ru/msb9134}
\crossref{https://doi.org/10.4213/sm9134}
\elib{http://elibrary.ru/item.asp?id=38593083}
\transl
\jour Sb. Math.
\yr 2019
\vol 210
\issue 8
\pages 1179--1205
\crossref{https://doi.org/10.1070/SM9134}


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  • https://doi.org/10.4213/sm9134
  • http://mi.mathnet.ru/eng/msb/v210/i8/p120

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