Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T02:32:16.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct derivations of certain surface integral formulae for the mean projections of a convex set

Published online by Cambridge University Press:  01 July 2016

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

Abstract

A simple direct proof is given of Minkowski's result that the mean length of the orthogonal projection of a convex set in E3 onto an isotropic random line is (2π)–1 times the integral of mean curvature over its surface. This proof is generalised to a correspondingly direct derivation of an analogous formula for the mean projection of a convex set in En onto an isotropic random s-dimensional subspace in En. (The standard derivation of this, and a companion formula, to be found in Bonnesen and Fenchel's classic book on convex sets, is most indirect.) Finally, an alternative short inductive derivation (due to Matheron) of both formulae, by way of Steiner's formula, is presented.

Keywords

GEOMETRICAL PROBABILITYINTEGRAL GEOMETRYSTEREOLOGYISOTROPIC RANDOM SUBSPACEMEAN PROJECTIONQUERMASSINTEGRALCONVEX SETMIXED VOLUMESURFACE INTEGRAL
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 4 , December 1975 , pp. 818 - 829
Copyright
Copyright © Applied Probability Trust 1975 

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] Bonnesen, T. and Fenchel, W. (1948) Theorie der konvexen Körper. Chelsea, New York.Google Scholar
[2] Busemann, H. (1958) Convex Surfaces. Wiley, New York.Google Scholar
[3] Cahn, J. W. (1967) The significance of average mean curvature and its determination by quantitative metallography. Trans. Met. Soc. AIME 239, 610–616.Google Scholar
[4] Dehoff, R. T. (1967) The quantitative estimation of mean surface curvature. Trans. Met. Soc. AIME 239, 617–621.Google Scholar
[5] Dehoff, R. T. and Rhines, F. N. (eds.) (1968) Quantitative Microscopy. McGraw-Hill, New York.Google Scholar
[6] Deltheil, R. (1926) Probabilités Géométriques. Gauthier-Villars, Paris.Google Scholar
[7] Firey, W. J. (1974) Extension of some integral formulas of Chern. Geometriae Dedicata 3, 325–333.CrossRefGoogle Scholar
[8] Hadwiger, H. (1957) Vorlesungen. über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin.Google Scholar
[9] Kendall, M. G. and Moran, P.A.P. (1963) Geometrical Probability. Hafner, New York.Google Scholar
[10] Miles, R. E. (1969) Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats. Adv. App. Prob. 1, 211–237.Google Scholar
[11] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
[12] Minkowski, H. (1903) Volumen und Oberfläche. Math. Ann. 57, 447–495.CrossRefGoogle Scholar
[13] Struik, D. J. (1961) Lectures on Classical Differential Geometry. Addison-Wesley, Mass. Google Scholar
[14] Underwood, E. E. (1970) Quantitative Stereology. Addison-Wesley, Mass. Google Scholar
[15] Weibel, E. R., Meek, G., Ralph, B., Echlin, P. and Ross, R. (eds.) Stereology 3 (Proceedings of the Third International Congress for Stereology, Berne, 1971). Blackwell, Oxford (also published in J. Micr. 95 (1972)).Google Scholar