  We study extreme points of the set of finite-outcome positive-operator-valued
measures (POVMs) on finite-dimensional Hilbert spaces and particularly the
possible ranks of the effects of an extreme POVM. We give results discussing
ways of deducing new rank combinations of extreme POVMs from rank combinations
of known extreme POVMs and, using these results, show ways to characterize rank
combinations of extreme POVMs in low dimensions. We show that, when a rank
combination together with a given dimension of the Hilbert space solve a
particular packing problem, there is an extreme POVM on the Hilbert space with
the given ranks. This geometric method is particularly effective for
constructing extreme POVMs with desired rank combinations.
