On smallest triangles
| dc.creator | Grimmett, Geoffrey | |
| dc.creator | Janson, Svante | |
| dc.date | 2002-08-06 | |
| dc.date.accessioned | 2026-07-07T04:50:04Z | |
| dc.date.available | 2026-07-07T04:50:04Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/math/0208044 | |
| dc.identifier | http://arxiv.org/abs/math/0208044 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64667 | |
| dc.subject | Probability | |
| dc.subject | 60D05, 60G55, 05C80, 60F05 | |
| dc.title | On smallest triangles | |
| dc.type | text |