Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-25T19:00:17.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Isotropic random simplices

Published online by Cambridge University Press:  01 July 2016

R. E. Miles
R. E. Miles*
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

Some obscure, yet fundamental, formulae of integral geometry are re-considered. They are applied to determine all the moments of the random volume of various isotropic random r-dimensional simplices in En (r = 1, …, n).

Type
Research Article
Information
Advances in Applied Probability , Volume 3 , Issue 2 , Autumn 1971 , pp. 353 - 382
Copyright
Copyright © Applied Probability Trust 1971 

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] Anderson, T. W. (1958) An Introduction to Multivariate Statistical Analysis. Wiley, New York.Google Scholar
[2] Birkhoff, G. and Maclane, S. (1953) A Survey of Modern Algebra. Macmillan, New York (Revised Edn).Google Scholar
[3] Blaschke, W. (1935) Integralgeometrie 1. Ermittlung der Dichten für lineare Unterraume im E n . Hermann, Paris (Act. Sci. Indust. No. 252).Google Scholar
[4] Blaschke, W. (1935) Integralgeometrie 2. Zu Ergebnissen von M. W. Crofton. Bull. Math. Soc. Roumaine des Sci. 37, 3–11.Google Scholar
[5] Busemann, H. (1953) Volume in terms of concurrent cross-sections. Pacific J. Math. 3, 1–12.Google Scholar
[6] Deltheil, R. (1926) Probabilités Géométriques. Gauthier-Villars, Paris.Google Scholar
[7] Flanders, H. (1963) Differential Forms with Applications to the Physical Sciences. Academic Press, New York (Monographs on Mathematics in Science and Engineering, Number 11).Google Scholar
[8] Hammersley, J. M. (1950) The distribution of distance in a hypersphere. Ann. Math. Statist. 21, 447–452.CrossRefGoogle Scholar
[9] Hostinsky, B. (1925) Sur les probabilités géométriques. Publ. Fac. Sci. Univ. Masaryk Brno, 3–26.Google Scholar
[10] James, A. T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 40–75.Google Scholar
[11] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Hafner, New York.Google Scholar
[12] Kingman, J. F. C. (1969) Random secants of a convex body. J. Appl. Prob. 6, 660–672.CrossRefGoogle Scholar
[13] Lord, R. D. (1954) The distribution of distance in a hypersphere. Ann. Math. Statist. 25, 794–798.CrossRefGoogle Scholar
[14] Miles, R. E. (1969) Probability distribution of a network of triangles (a solution to problem 67–15). SIAM Rev. 11, 399–402.CrossRefGoogle Scholar
[15] 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
[16] Miles, R. E. (1970) On the homogeneous planar Poisson point process. Math. Biosciences 6, 85–127.Google Scholar
[17] Miles, R. E. (1970) A synopsis of ‘Poisson flats in Euclidean spaces’. Izv. Akad. Nauk. Armen. SSR (Matematika) 5, 263–285.Google Scholar
[18] Miles, R. E. (1971) Random points, sets and tessellations on the surface of a sphere, Sankhyā A 33 (to appear).Google Scholar
[19] Moran, P. A. P. (1968) An Introduction to Probability Theory. Oxford Univ. Press.Google Scholar
[20] Nachbin, L. (1965) The Haar Integral. Van Nostrand, Princeton.Google Scholar
[21] Pearson, K. (1968) Tables of the Incomplete Beta-Function. Cambridge Univ. Press. 2nd Edition.Google Scholar
[22] Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum. Abh. Math. Seminar Hamburg 11, 249–310.CrossRefGoogle Scholar
[23] Raynaud, H. (1970) Sur l'enveloppe convexe des nuages de points aléatoires dans R n . I. J. Appl. Prob. 7, 35–48.Google Scholar
[24] Santaló, L. A. (1953) Introduction to Integral Geometry. Hermann, Paris (Act. Sci. Indust. No. 1198).Google Scholar
[25] Santaló, L. A. (1955) Sur la mesure des espaces linéaires qui coupent un corps convexe et problèmes qui s'y rattachent. Colloque sur les questions de réalité en géométrie, Liège, 177–190. Georges Thone, Liège; Masson et Cie, Paris.Google Scholar
[26] Slater, L. J. (1966) Generalized Hypergeometric Functions. Cambridge University Press.Google Scholar
[27] Stoka, M. I. (1967) Geometrie Integrală. Editura Academiei Republicii Socialiste Romania.Google Scholar
[28] Varga, O. (1935) Integralgeometrie 3. Croftons Formeln für den Raum. Math. Z 40, 387–405.Google Scholar