Given a homogeneous Poisson process on R^d with intensity L, we prove that it
is possible to partition the points into two sets, as a deterministic function
of the process, and in an isometry-equivariant way, so that each set of points
forms a homogeneous Poisson process, with any given pair of intensities summing
to L. In particular, this answers a question of Ball, who proved that in d=1,
the Poisson points may be similarly partitioned (via a translation-equivariant
function) so that one set forms a Poisson process of lower intensity, and asked
whether the same was possible for all d.