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1 results

  • The probability that two samples in the plane will have disjoint convex hulls

  • L. C. G. Rogers
  • Journal:
    Journal of Applied Probability / Volume 15 / Issue 4 / December 1978
    Published online by Cambridge University Press:
    14 July 2016, pp. 790-802
    Print publication:
    December 1978

  • Suppose given an absolutely continuous distribution on the plane, and points Pi, · · ·, Pj, Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {Pi, · · ·, Pj}, and by Γk the convex hull of (Πt, · · ·, Πk}, and writing pjk for the probability that Gj and Γk are disjoint, certain properties of the array {pjk; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p1n; n = 1,2, · · ·}, and asymptotic results for {pnn; n = 1,2, · · ·}. Some examples are considered.