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Poisson flats in Euclidean spaces Part I: A finite number of random uniform flats

Published online by Cambridge University Press:  01 July 2016

R. E. Miles*
Affiliation:
Australian National University

Extract

This is the first part of a three part work, the common setting being Ed, Euclidean space of d dimensions. Many random physical phenomena, often in the form of structures, admit models which are assemblages of random s-flats in Ed. Indeed, taking s = 0, any n-sample from a d-dimensional distribution may of course be so regarded! For static phenomena generally 0 ≦ s < d ≦ 3, while 4 is to be substituted for 3 if time variation is allowed. By postulating a high degree of stochastic independence and uniformity, a variety of “simple” models is defined. However, their investigation poses problems in geometrical probability of a wide range of difficulty; the solutions of many of which are still far from complete.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 

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References

[1] Birkhoff, G and Maclane, S. (1953) A Survey of Modern Algebra. Macmillan, New York, Revised Edition.Google Scholar
[2] Blaschke, W. (1935) Integralgeometrie 1. Ermittlung der Dichten für lineare Unterraume im E n . Hermann, Paris (Act. Sci. Indust. No. 252).Google Scholar
[3] Blaschke, W. (1949) Vorlesungen über Integralgeometrie. Chelsea, New York.Google Scholar
[4] Bonnesen, T. and Fenchel, W. (1948) Theorie der konvexen Körper. Chelsea, New York.Google Scholar
[5] Busemann, H. (1958) Convex Surfaces. Wiley, New York.Google Scholar
[6] Cesari, L. (1956) Surface Area. Princeton U. P. (Ann. Math. Studies No. 35).Google Scholar
[7] Crofton, M. W. (1868) On the theory of local probability, applied to straight lines drawn at random in a plane; … Philos. Trans. Roy. Soc. 158, 181199.Google Scholar
[8] Crofton, M. W. (1869) Sur quelques théorèmes de calcul intègral. C. R. Acad. Sci. Paris, 68, 14691470.Google Scholar
[9] Deltheil, R. (1926) Probabilités Géométriques. Gauthier-Villars, Paris.Google Scholar
[10] Eggleston, H. G. (1958) Convexity. Cambridge U. P. CrossRefGoogle Scholar
[11] grünbaum, B. (1967) Convex Polytopes. Wiley, New York.Google Scholar
[12] Hadwiger, H. (1950) Neue Integralrelationen für Eikörperpaare. Acta Sci. Math. 13, 252257.Google Scholar
[13] Hadwiger, H. (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag. Google Scholar
[14] Hostinsky, B. (1925) Sur les probabilités géométriques. Publ. Fac. Sci. Univ. Masaryk, Brno, 326.Google Scholar
[15] James, A. T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 4075.CrossRefGoogle Scholar
[16] Kendall, M. G. (1961) A Course in the Geometry of n Dimensions. Hafner, New York.Google Scholar
[17] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Hafner, New York.Google Scholar
[18] Miles, R. E. (1961) Random Polytopes: The Generalisation to n Dimensions of the Intervals of a Poisson process. , Cambridge University.Google Scholar
[19] Miles, R. E. (1964) A wide class of distributions in geometrical probability (abstract). Ann. Math. Statist. 35, 1407.Google Scholar
[20] Miles, R. E. (1964) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. 52, 901907, II. 1157–1160.CrossRefGoogle ScholarPubMed
[21] Miles, R. E. (1965) Contribution to the discussion of Professor Pyke's paper. J. R. Statist. Soc. B 27, 444.Google Scholar
[22] Miles, R. E. On the homogeneous planar Poisson point process. Mathematical Biosciences (to appear).Google Scholar
[23] Santaló, L. A. (1952) Integral geometry in spaces of constant curvature (Spanish. English summary.) Repub. Argentina Publ. Com. Nac. Energia Atomica Ser. Mat. 1, no. 1.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, 177190. Georges Thone, Liège; Masson et Cie, Paris.Google Scholar
[26] Santaló, L. A. (1956) On mean curvatures of a flattened convex body. Rev. Fac. Sci. Univ. Istanbul Ser. A 21, 189194.Google Scholar
[27] Sommerville, D. M. Y. (1958) An Introduction to the Geometry of N Dimensions. Dover, New York.Google Scholar
[28] Stoka, M. I. (1967) Geometrie Integrala. Editura Academiei Republicii Socialiste Romania.Google Scholar
[29] Uzawa, Hirofumi (1958) A theorem on convex polyhedral cones. Studies in Linear and Non-linear Programming by Arrow, Hurwicz, Uzawa et alia. Stanford U. P., California, 2331.Google Scholar