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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.
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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 

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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