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.
"Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points." Ann. Probab. 36 (1) 363 - 396, January 2008. https://doi.org/10.1214/009117907000000259