Geodesic
An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 223-236

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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. 
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     author = {V. A. Myrikova},
     title = {An {Isoperimetric} {Problem} on the {Lobachevsky} {Plane} with a {Left-Invariant} {Finsler} {Structure}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {223--236},
     publisher = {mathdoc},
     volume = {321},
     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/TM_2023_321_a14/}
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V. A. Myrikova. An Isoperimetric Problem on the Lobachevsky Plane with a Left-Invariant Finsler Structure. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 223-236. http://geodesic.mathdoc.fr/item/TM_2023_321_a14/