Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T09:27:07.159Z Has data issue: false hasContentIssue false
  • English
  • Français

Random triangles in convex regions

Published online by Cambridge University Press:  14 July 2016

Norbert Henze
Norbert Henze*
Affiliation:
University of Hannover
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Hannover, Welfengarten 1, D-3000 Hannover 1, Federal Republic of Germany.
Get accessRights & Permissions [Opens in a new window]

Abstract

Consider a random triangle formed by vertices which are independent and distributed uniformly inside a given bounded convex region K. Then a well-known result of W. Blaschke states that, among all regions of fixed area, the area AK of this random triangle is stochastically bounded by A circle (A c) and A triangle (A t). The distribution of the latter was obtained by Alagar (1977). We use the classical Crofton technique to derive an integral representation for the distribution function F c of A c from which F c is calculated using numerical quadrature. The maximum deviation between Fc and the distribution function of A t is less than 0.04. Additionally, we obtain an explicit form for the distribution of A parallelogram which may be represented as a mixture of functions involving Dirichlet-distributed random vectors.

Keywords

RANDOM TRIANGLEGEOMETRIC PROBABILITY

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983, pp. 111 - 125
Copyright
Copyright © Applied Probability Trust 1983 

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.)

Article purchase

Temporarily unavailable

References

Alagar, V. S. (1977) On the distribution of a random triangle. J. Appl. Prob. 14, 284–297.Google Scholar
Biehl, T. (1923) Affine Geometrie XXXVIII: Über die Schüttlung von Eikörpern. Abh. Math. Sem. Hamburg 2, 69–70.Google Scholar
Blaschke, W. (1917) Affine Geometrie XI: Lösung des ‘Vierpunktproblems’ von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Ber. Verh. Sächs. Ges. Wiss. Leipzig 69, 436–653.Google Scholar
Blaschke, W. (1923) Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie. Springer-Verlag, Berlin.Google Scholar
Groemer, H. (1973) On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math. 45, 525–533.Google Scholar
Groemer, H. (1974) On the mean value of the volume of a random polytope in a convex set. Arch. Mat. (Basel) 25, 86–90.Google Scholar
Kendall, M. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Klee, V. (1969) What is the expected volume of a simplex whose vertices are chosen at random from a given convex body? Amer. Math. Monthly 76, 286–288.CrossRefGoogle Scholar
Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
Reed, W. J. (1974) Random points in a simplex. Pacific J. Math. 54, 183–198.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Ma.Google Scholar