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Acute triangles in the n-ball

Published online by Cambridge University Press:  14 July 2016

Glen Richard Hall
Glen Richard Hall*
Affiliation:
University of Minnesota
*
∗Postal address: School of Mathematics, University of Minnesota Twin Cities, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, U.S.A.
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Abstract

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

Keywords

GEOMETRIC PROBABILITYCROFTON'S DIFFERENTIAL EQUATIONRANDOM TRIANGLESMOVING MANIFOLDS
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 712 - 715
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Baddeley, A. (1977) Integrals on a moving manifold and geometrical probability. Adv. Appl. Prob. 9, 588–603.Google Scholar
[2] Baddeley, A. (1977) A fourth note on recent research in geometrical probability. Adv. Appl. Prob. 9, 824–860.Google Scholar
[3] Broadbent, S. R. (1980) Simulating the ley hunters. J. R. Statist. Soc. A 143, 109–140.Google Scholar
[4] Kendall, D. G. and Kendall, W. S. (1980) Alignment in two-dimensional random point sets. Adv. Appl. Prob. 12, 380–424.CrossRefGoogle Scholar
[5] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
[6] Langford, E. (1969) Probability that a random triangle is obtuse. Biometrika 56, 689–690.Google Scholar