The Isoperimetric Problem In Randers Planes

TL;DR

This paper investigates the isoperimetric problem in Randers planes, showing that origin-centered circles maximize area locally under certain volume forms, extending classical Euclidean results to Finsler geometry.

Contribution

It demonstrates that in Randers planes, circles centered at the origin locally maximize area, providing new insights into isoperimetric solutions in Finsler geometry.

Findings

Origin-centered circles achieve local maximum area in Randers planes.

The result extends classical isoperimetric principles to Finsler geometry.

The study clarifies the role of volume forms in the isoperimetric problem.

Abstract

In this paper, the isoperimetric problem in Randers planes, $(\mathbb{R}^2,F=\alpha +\beta)$, which are slight deformation of the Euclidean plane $(\mathbb{R}^2,\alpha)$ by suitable one forms $\beta$, have been studied. We prove that the circles centred at the origin achieves the local maximum area of the isoperimetric problem with respect to well known volume forms in Finsler geometry.