The probability that two samples in the plane will have disjoint convex hulls
- 1 March 1978
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 15 (04) , 790-802
- https://doi.org/10.1017/s0021900200026139
Abstract
Suppose given an absolutely continuous distribution on the plane, and points P i, · · ·, Pj , Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {P i, · · ·, Pj }, and by Γk the convex hull of (Πt, · · ·, Π k }, and writing pjk for the probability that Gj and Γ k are disjoint, certain properties of the array {pjk ; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p 1n ; n = 1,2, · · ·}, and asymptotic results for {pnn ; n = 1,2, · · ·}. Some examples are considered.Keywords
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