Spherical isoperimetric inequality

A simple closed curve of length $L$ on the unit sphere enclosing the smaller area $A$ satisfiesEquality holds for a circle. A variational proof uses Gauss--Bonnet to turn area maximization at fixed length into geodesic-curvature minimization, proves that an extremizer has constant geodesic curvature and is planar, and evaluates its spherical cap.

On the unit sphere, write $x=A(V)$ and $s=4\pi$. The expression equals $|xA(D\setminus V)-(s-x)A(D\cap V)|$, at most $x(s-x)$. A constant-version spherical isoperimetric inequality therefore gives the displayed bound, uniformly in nonempty open $D$. A smooth positive area density $\theta$ gives a second metric; integrating this bound over $D_t=\{\theta >t\}$ by the layer cake representation bounds the discrepancy between the two normalized area fractions by $\kappa (\max \theta -\min \theta )L^2/(s\tilde{A}(S))$.