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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Volume 321, Pages 223–236
DOI: https://doi.org/10.4213/tm4325 (Mi tm4325) 		 
An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure

V. A. Myrikova

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
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DOI: https://doi.org/10.4213/tm4325
Abstract: A Finsler analog of the Lobachevsky plane is the Lie group of proper affine transformations of the real line with a left-invariant Finsler structure generated by a convex compact set in the Lie algebra with the origin in its interior. We consider the isoperimetric problem on this Lie group, with the volume form also taken to be left-invariant. This problem is formulated as an optimal control problem. Applying the Pontryagin maximum principle, we find the optimal isoperimetric loops in an explicit form in terms of convex trigonometry functions. We also present a generalized isoperimetric inequality in a parametric form.
Keywords: Finsler geometry, isoperimetric problem, isoperimetric inequality, Lobachevsky plane, hyperbolic plane, optimal control, convex trigonometry.
Funding agency Grant number
Russian Science Foundation 20-11-20169
This work was supported by the Russian Science Foundation under grant no. 20-11-20169, https://rscf.ru/en/project/20-11-20169/.
Received: April 12, 2022
Revised: July 30, 2022
Accepted: January 9, 2023
English version:
Proceedings of the Steklov Institute of Mathematics, 2023, Volume 321, Pages 208–221
DOI: https://doi.org/10.1134/S0081543823020153
Bibliographic databases:
Document Type: Article
UDC: 517.977
Language: Russian
Citation: V. A. Myrikova, “An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 223–236; Proc. Steklov Inst. Math., 321 (2023), 208–221
Citation in format AMSBIB
\Bibitem{Myr23}
\by V.~A.~Myrikova
\paper An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure
\inbook Optimal Control and Dynamical Systems
\bookinfo Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 321
\pages 223--236
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4325}
\crossref{https://doi.org/10.4213/tm4325}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4643643}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 321
\pages 208--221
\crossref{https://doi.org/10.1134/S0081543823020153}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85170851207}
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