Convex polygons and the isoperimetric problem in simply connected space forms $M_{\kappa}^2$

TL;DR

This paper proves the uniqueness of perimeter-minimizing circles for given areas in simply connected space forms and establishes the isoperimetric inequality using a uniform geometric approach.

Contribution

It demonstrates the existence and uniqueness of isoperimetric circles in all simply connected space forms with a unified geometric proof.

Findings

Unique perimeter minimizer is a circle in $M_{ ext{kappa}}^2$

Explicit formula for the radius of the minimizing circle

Established the isoperimetric inequality in all three space forms

Abstract

In this article, we prove that there exists a unique perimeter minimizer among all piecewise smooth simple closed curves in $M_{\kappa}^2$ enclosing area $A > 0$ $(A \leq 2{\pi}$ if ${\kappa} = 1)$, and it is a circle in $M_{\kappa}^2$ of radius $AS_{\kappa} \left( \dfrac{ \sqrt{ A ( 4 {\pi} - {\kappa} A ) } }{ 2 {\pi} } \right)$, where $AS_{\kappa}(t) := t$ if ${\kappa} = 0$, arcsin$(t)$ if ${\kappa} = 1$, sinh$^{-1}(t)$ if ${\kappa} =-1$. We also prove the isoperimetric inequality for $M_{\kappa}^2$. We give an elementary geometric proof which is uniform for all three simply connected space forms.