  Pick n points independently at random in R^2, according to a prescribed
probability measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the
binomial n choose 3 triangles thus formed, in non-decreasing order. If mu is
absolutely continuous with respect to Lebesgue measure, then, under weak
conditions, the set {n^3 D^n_i : i >= 1} converges as n --> infinity to a
Poisson process with a constant intensity c(mu). 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(mu) <= 2/|S|, where |S| denotes the area of S. Equality holds in that
c(mu) = 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.
