Convex polygons and the isoperimetric problem in simply connected space forms M_²

In this article, we prove that there exists a unique perimeter minimizer among all piecewise smooth simple closed curves in M_² enclosing area A > 0 (A 2 if = 1), and it is a circle in M_² of radius AS_ (A (4 { - { A) } } 2 { }), where AS_ (t): = t if = 0, arcsin (t) if = 1, sinh^-1 (t) if =-1. We also prove the isoperimetric inequality for M_². We give an elementary geometric proof which is uniform for all three simply connected space forms.