In various stereological problems an $n$-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.

翻译：在各种体视学问题中，一个$n$维凸体被一个$(n-1)$维各向同性均匀随机超平面所截。本文研究了与此类随机截面$(n-1)$维体积相关的累积分布函数。该分布在平面情形和空间情形下分别被称为弦长分布和截面面积分布。针对多类凸体，本文证明了这些分布函数关于勒贝格测度是绝对连续的。同时提出了一种用于逼近相应概率密度函数的蒙特卡洛模拟方案。