A Stable Marriage of Poisson and Lebesgue

Let be a discrete set in R d . Call the elements of centers. The well-known Voronoi tessellation partitions R d into polyhedral regions (of varying sizes) by allocating each site of R d to the closest center. Here we study \fair" allocations of R d to in which the regions allocated to dieren t centers have equal volumes. We prove that if is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale-Shapley marriage problem. (That is, sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center is a union of nitely many bounded connected sets. However, in the case of a Poisson process, an innite volume of sites are allocated to centers further away than . We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in d > 1.