Gravitational allocation for uniform points on the sphere

Given a collection $mathcal L$ of $n$ points on a sphere $mathbf{S}^2_n$ of surface area $n$, a fair allocation is a partition of the sphere into $n$ parts each of area $1$, and each associated with a distinct point of $mathcal L$. We show that if the $n$ points are chosen uniformly at random and the partition is defined by considering th...More