Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T14:56:36.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Some new integral geometric formulae, with stochastic applications

Published online by Cambridge University Press:  14 July 2016

R. E. Miles
R. E. Miles*
Affiliation:
The Australian National University
*
∗ Postal address: Research School of Social Sciences, Department of Statistics, IAS, The Australian National University, P.O.Box 4, Canberra, ACT 2600, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Alternative forms of the integral geometric density of an r-subspace [r-flat] containing q[q + 1] points in euclidean n-space Rn are given Stochastic applications in R3 include formulae for

(i) the mean area of intersection of a domain by an isotropic plane through the origin; and

(ii) the variance of the area of intersection of a domain by an isotropic uniform plane.

Keywords

GEOMETRICAL PROBABILITYINTEGRAL GEOMETRYINVARIANT DENSITYBLASCHKE–PETKANTSCHIN FORMULAISOTROPIC RANDOM SUBSPACEISOTROPIC UNIFORM RANDOM FLATMEAN PROJECTIONRANDOM SECANTCONVEX SETCROFTON'S FORMULARANDOM SIMPLEX
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 592 - 606
Copyright
Copyright © Applied Probability Trust 1979 

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] Anderssen, R. S., Brent, R. P., Daley, D. J. and Moran, P. A. P. (1976) Concerning and a Taylor series method. SIAM J. Appl. Math. 30, 22–30.Google Scholar
[2] Blaschke, W. (1935) Integralgeometrie 1. Ermittlung der Dichten für lineare Unterräume im En. Hermann, Paris (Act. Sci. Indust. 252).Google Scholar
[3] Blaschke, W. (1935) Integralgeometrie 2. Zu Ergebnissen von M. W. Crofton. Bull. Math. Soc. Roumaine Sci. 37, 3–11.Google Scholar
[4] Busemann, H. (1953) Volume in terms of concurrent cross-sections. Pacific J. Math. 3, 1–12.Google Scholar
[5] Coleman, R. (1969) Random paths through convex bodies. J. Appl. Prob. 6, 430–441.Google Scholar
[6] Coleman, R. (1973) Random paths through rectangles and cubes. Metallography 6, 103–114.Google Scholar
[7] Crofton, M. W. (1869) Sur quelques théorèmes de calcul intégral. C. R. Acad. Sci. Paris 68, 1469–1470.Google Scholar
[8] Enns, E. G. and Ehlers, P. F. (1978) Random paths through a convex region. J. Appl. Prob. 15, 144–152.Google Scholar
[9] Furstenberg, H. and Tzkoni, I. (1971) Spherical functions and integral geometry. Israel J. Math. 10, 327–338.Google Scholar
[10] Guggenheimer, H. (1973) A formula of Furstenberg–Tzkoni type. Israel I. Math. 14, 281–282.Google Scholar
[11] Hadwiger, H. (1950) Neue Integralrelationen für Eikörperpaare. Acta Sci. Math. 13, 252–257.Google Scholar
[12] Horowitz, M. (1965) Probability of random paths across elementary geometrical shapes. J. Appl. Prob. 2, 169–177.Google Scholar
[13] Hostinsky, B. (1925) Sur les probabilités géométriques. Publ. Fac. Sci. Univ. Masaryk Brno, 3–26.Google Scholar
[14] Itoh, H. (1970) An analytical expression of the intercept length distribution of cubic particles. Metallography 3, 407–419.Google Scholar
[15] Kellerer, A. M. (1971) Considerations of the random transversal of convex bodies and solutions for general cylinders. Radiation Res. 47, 359–376.Google Scholar
[16] Kingman, J. F. C. (1969) Random secants of a convex body. J. Appl. Prob. 6, 660–672.Google Scholar
[17] Miles, R. E. (1969) Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237.Google Scholar
[18] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
[19] Miles, R. E. (1973) A simple derivation of a formula of Furstenberg and Tzkoni. Israel J. Math. 14, 278–280.Google Scholar
[20] Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n -dimensionalen Raum. Abh. Math. Seminar Hamburg 11, 249–310.Google Scholar
[21] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. (Encyclopedia of Mathematics and its Applications, Vol. 1) Addison-Wesley, Reading, Mass.Google Scholar