Existence and Approximation of Densities of Chord Length- and Cross Section Area Distributions
Keywords:absolute continuity, chord length distribution, cross section area distribution, stereology
In various stereological problems a n-dimensional convex body is intersected with an (n−1)-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the (n−1)-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.
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Copyright (c) 2023 Thomas van der Jagt, Geurt Jongbloed, Martina Vittorietti
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