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Rotational versions of the Crofton formula

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Eva B. Vedel Jensen
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
*
* Postal address: Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark.
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Abstract

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.

Keywords

MINKOWSKI FUNCTIONALSHYPERGEOMETRIC FUNCTIONSLOCAL STEREOLOGY

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 87 - 96
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain on 21–24 September 1993.

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