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Inequalities for random flats meeting a convex body

Published online by Cambridge University Press:  14 July 2016

Rolf Schneider
Rolf Schneider*
Affiliation:
Universität Freiburg I. Br.
*
∗Postal address: Mathematisches Institut, Albert-Ludwigs-Universität, Albertstrasse 23b, D-7800 Freiburg i. Br., W. Germany.
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Abstract

We choose a uniform random point in a given convex body K in n-dimensional Euclidean space and through that point the secant of K with random direction chosen independently and isotropically. Given the volume of K, the expectation of the length of the resulting random secant of K was conjectured by Enns and Ehlers [5] to be maximal if K is a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meeting K, and we prove that certain geometric probabilities connected with these again become maximal when K is a ball.

Keywords

GEOMETRICAL PROBABILITYRANDOM SECANTS
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 3 , September 1985 , pp. 710 - 716
Copyright
Copyright © Applied Probability Trust 1985 

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References

[1] Busemann, H. (1958) Convex Surfaces. Interscience, New York.Google Scholar
[2] Carleman, T. (1919) Über eine isoperimetrische Aufgabe und ihre physikalischen Anwendungen. Math. Z. 3, 1–7.Google Scholar
[3] Coleman, R. (1969) Random paths through convex bodies. J. Appl. Prob. 6, 430–441.CrossRefGoogle Scholar
[4] Davy, P. and Miles, R. E. (1977) Sampling theory for opaque spatial specimens. J. R. Statist. Soc. B 39, 56–65.Google Scholar
[5] Enns, E. G. and Ehlers, P. F. (1978) Random paths through a convex region. J. Appl. Prob. 15, 144–152.CrossRefGoogle Scholar
[6] Groemer, H. (1973) On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math. 45, 525–533.CrossRefGoogle Scholar
[7] Kingman, J. F. C. (1965) Mean free paths in a convex reflecting region. J. Appl. Prob. 2, 162–168.Google Scholar
[8] Kingman, J. F. C. (1969) Random secants of a convex body. J. Appl. Prob. 6, 660–672.Google Scholar
[9] Knothe, H. (1937) Über Ungleichungen bei Sehnenpotenzintegralen. Deutsche Math. 2, 544–551.Google Scholar
[10] 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
[11] Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.Google Scholar
[12] Miles, R. E. and Davy, P. (1976) Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae. J. Microscopy 107, 211–226.Google Scholar
[13] Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum. Abh. Math. Sem. Univ. Hamburg 11, 249–310.Google Scholar
[14] Pfiefer, R. E. (1982) The Extrema of Geometric Mean Values. , University of California, Davis.Google Scholar
[15] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Mass.Google Scholar
[16] Schneider, R. (1982) Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. 61, 379–387.Google Scholar