Convex polygons and the isoperimetric problem in simply connected space forms Mκ²

In this article, we prove that there exists a unique perimeter minimizer among all piecewise smooth simple closed curves in Mκ² enclosing area $A > 0$ (A ≤ 2π if κ = 1), and it is a circle in Mκ² of radius ASκ ( √ A ( 4 π - κ A ) 2 π ), where ASκ(t) := t if κ = 0, arcsin$(t)$ if κ = 1, sinh⁻¹(t) if κ =-1. We also prove the isoperimetric inequality for Mκ². We give an elementary geometric proof which is uniform for all three simply connected space forms.