  Let $\Xi$ be a discrete set in ${\mathbb{R}}^d$. Call the elements of $\Xi$
centers. The well-known Voronoi tessellation partitions ${\mathbb{R}}^d$ into
polyhedral regions (of varying sizes) by allocating each site of
${\mathbb{R}}^d$ to the closest center. Here we study ``fair'' allocations of
${\mathbb{R}}^d$ to $\Xi$ in which the regions allocated to different centers
have equal volumes. We prove that if $\Xi$ 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. (I.e.,
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 $\xi$ is a union of finitely many bounded connected sets. However,
in the case of a Poisson process, an infinite volume of sites are allocated to
centers further away than $\xi$. 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 $d>1$.
