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Random circles in the d-dimensional unit ball

Published online by Cambridge University Press:  14 July 2016

Fernando Affentranger
Fernando Affentranger*
Affiliation:
Universidad de Buenos Aires
*
∗ Present address: Mathematisches Institut, Albert-Ludwigs-Universität, Albertstrasse 23B, D-7800 Freiburg im Br., W. Germany.
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Abstract

This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

Keywords

GEOMETRY OF THE d-BALLGEOMETRICAL PROBABILITYINTEGRAL GEOMETRYRANDOM SET OF POINTSSTOCHASTIC GEOMETRY
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 408 - 412
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Buchta, C. (1986) A note on the volume of a random polytope in a tetrahedron. Illinois J. Math. 30, 653–659.CrossRefGoogle Scholar
[2] Buchta, C. (1986) On a conjecture of R. E. Miles about the convex hull of random points. Mh. Math. 102, 91–102.CrossRefGoogle Scholar
[3] Deltheil, R. (1926) Probabilitiés Géométriques. Traité du calcul des probabilités et de ses applications. Gauthier-Villars, Paris.Google Scholar
[4] Hall, G. R. (1982) Acute triangles in the n-ball. J. Appl. Prob. 19, 712–715.CrossRefGoogle Scholar
[5] Hammersley, J. M. (1950) The distribution of distance in a hypersphere. Ann. Math. Statist. 21, 447–452.CrossRefGoogle Scholar
[6] Lord, R. D. (1954) The distribution of distance in a hypersphere. Ann. Math. Statist. 25, 794–798.CrossRefGoogle Scholar
[7] Miles, R. E. (1970) On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.CrossRefGoogle Scholar
[8] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
[9] Petkantschin, B. (1936) Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raume. Abh. Math. Sem. Univ. Hamburg 11, 249–310.CrossRefGoogle Scholar
[10] Ruben, H. (1977) The volume of a random simplex in an n-ball is asymptotically normal. J. Appl. Prob. 14, 647–653.CrossRefGoogle Scholar
[11] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Massachusetts.Google Scholar