= Spherical inclusion area discrepancy
{title2=$|A(V)A(D)-A(S)A(V\cap D)|\le\kappa L(\partial V)^2$}
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 $\kappa(\max\theta-\min\theta)L^2/(s\widetilde A(S))$.
Back to article page
