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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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