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

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 $A>0$ $(A\le 2\pi$ if $\kappa =1)$, and it is a circle in $M_{\kappa }^2$ of radius $A{S}_{\kappa }\left(\frac{\sqrt{A(4\pi -\kappa A)}}{2\pi }\right)$, where $A{S}_{\kappa }(t):=t$ 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.

35 pages, 8 figures