Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T13:34:26.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Dual Problems of Geometric Probability in the Plane

Published online by Cambridge University Press:  12 September 2008

John Gates
John Gates
Affiliation:
Applied Statistics and Operational Modelling Section, School of Mathematics, Statistics and Computing, Greenwich University, Wellington Street, Woolwich, London SE18 6PF
Get access Rights & Permissions [Opens in a new window]

Abstract

A definition is adopted for convexity of a set of directed lines in the plane. Following this, the duals of a number of standard problems of geometric probability are formulated. Problems considered in detail are the duals of Sylvester's problem, chord length distributions and Ambartzumian's combinatorial geometry. The paper suggests some questions for further work.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 1 , March 1993 , pp. 11 - 23
Copyright
Copyright © Cambridge University Press 1993

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]Alagar, V. S. (1977) On the distribution of a random triangle. J. Appl. Prob. 14, 284–297.CrossRefGoogle Scholar
[2]Ambartzumian, R. V. (1982) Combinatorial Integral Geometry, Wiley, Chichester.Google Scholar
[3]Ambartzumian, R. V. (1990) Factorisation Calculus and Geometric Probability, Cambridge University Press, Cambridge.Google Scholar
[4]Baddeley, A. (1982) In: Ambartzumian, R. V. (Appendix A) Combinatorial Integral Geometry Wiley, Chichester.Google Scholar
[5]Carver, W. B. (1941) The polygonal regions into which a plane is divided by n straight lines. Amer. Math. Monthly 48 667–675.CrossRefGoogle Scholar
[6]Davidson, R. (1974) Stochastic processes of flats and exchangeability. In Harding, E. F. and Kendall, D. G. (eds.) Stochastic Geometry, Wiley, London.Google Scholar
[7]Gates, J. (1987) Some properties of chord length distributions. J. Appl. Prob. 24 863–873.CrossRefGoogle Scholar
[8]Kendall, D. G. (1974) An introduction to Stochastic Geometry. In Harding, E. F. and Kendall, D. G. (eds.) Stochastic Geometry, Wiley, London.Google Scholar
[9]Kendall, D. G. (1985) Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Prob. 17 308–329.CrossRefGoogle Scholar
[10]Santalo, L. A. (1976) Integral Geometry and Geometric Probability, Addison-Wesley, Massachusetts.Google Scholar