On an isoperimetric problem on the Lobachevsky hyperbolic plane with left-invariant Finsler structure

TL;DR

This paper investigates an isoperimetric problem on the Finsler hyperbolic plane modeled as a Lie group, deriving optimal loops using convex trigonometry and proposing a generalized inequality.

Contribution

It introduces a novel isoperimetric problem on a Finsler hyperbolic plane with explicit solutions and a parametric generalization of the inequality.

Findings

Optimal isoperimetric loops characterized by convex trigonometry functions

Generalized isoperimetric inequality in parametric form

Application to the Lie group of affine transformations

Abstract

We deal with an isoperimetric problem on the Finsler hyperbolic plane. The space is defined as the Lie group of proper affine transformations of the line with a left-invariant Finsler structure. To state the problem, we use the left-invariant volume form. The optimal isoperimetric loops are found in terms of convex trigonometry functions. We also propose a generalization of the isoperimetric inequality in parametric form.