On smallest triangles

Geoffrey Grimmett and Svante Janson Abstra t. Pick n points independently at random in R2 , according to a prescribed probability measure μ, and let ∆ 1 ≤ ∆ 2 ≤ . . . be the areas of the n 3 triangles thus formed, in non-decreasing order. If μ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n∆ i : i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(μ). 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 μ is the uniform probability measure on the region S, then κ(μ) ≤ 2/|S|, where |S| denotes the area of S. Equality holds in that κ(μ) = 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 Vitányi.