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