  We show that the random point measures induced by vertices in the convex hull
of a Poisson sample on the unit ball, when properly scaled and centered,
converge to those of a mean zero Gaussian field. We establish limiting variance
and covariance asymptotics in terms of the density of the Poisson sample.
Similar results hold for the point measures induced by the maximal points in a
Poisson sample. The approach involves introducing a generalized spatial birth
growth process allowing for cell overlap.
