An isoperimetric deficit formula (Q2024670)

Let \(\Omega\) be a bounded domain with \(C^1\) boundary in either \(\mathbb{R}^2,\mathbb{S}^2\) or \(\mathbb{H}^2\). Let \(K\) be the Gaussian curvature of the ambient space, let \(L\) be the lenght of \(\partial \Omega\) and let \(A\) be the area of \(\Omega\). The author shows that the isoperimetric deficit \(L^2-4\pi A+KA^2\) can be expressed as an integral comparing the outward normal field along \(\partial \Omega\) at two points. The formula is shown in the Euclidean case and then extended to the non-zero curvature case.