Abstract

Pick n points independently at random in R^2, according to a prescribed probability measure mu, and let D^n_1 = 1} converges as n --> infinity to a Poisson process with a constant intensity c. This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that, if mu is the uniform probability measure on the region S, then c <= 2/|S|, where |S| denotes the area of S. Equality holds in that c = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitanyi.