Hostname: page-component-7bb8b95d7b-fmk2r Total loading time: 0 Render date: 2024-09-07T03:21:31.322Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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