The variance of a poisson process of domains

A familiar relation links the densities that result for the
intersection of a convex body and straight lines under uniform
isotropic randomness with those that result under weighted
randomness. An extension of this relation to the intersection of
more general domains is utilized to obtain the variance of the
n-dimensional measure of the intersection of two bodies under
uniform isotropic randomness. The formula for the variance contains
the point-pair distance distributions for the two domains — or the
closely related geometric reduction factors. The result is applied
to derive the variance of the intersection of a Boolean scheme,
i.e. a stationary, isotropic Poisson process of domains, with a
fixed sampling region.