Relative isoperimetric inequality on a curved surface (Q2467632)

The author proves isoperimetric inequalities of the type \[ 2\pi \operatorname{Area}(\Omega) - K \operatorname{Area} (\Omega)^2 \leq \operatorname{Length} (\partial \Omega \sim \partial C)^2 \] for \(S\) being a complete simply connected surface of Gaussian curvature bounded above by the constant \(K\), \(C\) being a closed convex set in \(S\) and \(\Omega\) a relatively compact subset in \(S\sim C\). He also shows that equality holds if and only if \(\Omega\) is a geodesic half disk with constant Gaussian curvature and \(\partial \Omega \sim \partial C\) is a geodesic semicircle.