Existence and approximation of densities of chord length- and cross section area distributions

Abstract

In various stereological problems ann-dimensional convex body is intersected with an(n−1)-dimensionalIsotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associatedwith the(n−1)-dimensional volume of such a random section is studied. This distribution is also knownas 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 continuouswith respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating thecorresponding probability density functions.