AI 中文总结
本文推导了长方体在随机方向下宽度分布的闭式累积分布函数，分类了奇点并计算系数，得到各阶累积量的性质，还将方法扩展到中心对称体为zonotope的几何体及所有满维多面体。
AI 中文摘要
对于边为$a_1,a_2,a_3$的长方体，在单位球$S^2$中均匀随机方向$u$下的宽度为$w=\sum_i a_i|u_i|$。通过双射参数变换，该宽度也等于长方体的投影面积，其分布已由Walters推导得出。本文给出了直接的余面积推导，将该分布重铸为一个全局正部公式，并积分得到初等闭式累积分布函数。该密度函数除在边长、面对角线和空间对角线处外均为实解析函数，我们将每个奇点分类为角点、平方根折叠、叠加或终端跳跃，并计算其系数。高斯表示可从一个生成函数得到所有矩；第$n$阶累积量在$\pi^{-1}$中的次数恰好为$\lfloor n/2 \rfloor$，且各阶的均值归一化累积量是内蕴体积的两个尺度不变组合的多项式，直至五阶均给出了显式公式。前三个累积量可确定该长方体，我们还刻画了可允许的三元组。除长方体外，内蕴体积既不能确定宽度方差也不能确定亮度方差。符号模式表示可扩展到中心对称体为zonotope（zonotope：中心对称多面体）的几何体，而逐单元余面积方法适用于所有满维多面体。
英文摘要
For a rectangular box with edges $a_1,a_2,a_3$, the width in a uniform random direction $u \in S^2$ is $w=\sum_i a_i|u_i|$. Under a bijective change of parameters this is also the projected area of a rectangular parallelepiped, whose distribution was derived by Walters. We give a direct co-area derivation that recasts the law as one global positive-part formula and integrates it to an elementary closed cumulative distribution function. The density is real-analytic except at the edge lengths, face diagonals and space diagonal. We classify every singularity as a corner, square-root fold, superposition or terminal jump and compute its coefficient. A Gaussian representation gives all moments from one generating function. The degree of the $n$th cumulant in $π^{-1}$ is exactly $\lfloor n/2 \rfloor$; at every order the mean-normalised cumulant is a polynomial in two scale-invariant combinations of the intrinsic volumes, with explicit formulas given through fifth order. The first three cumulants determine the box, and we characterise the admissible triples. Beyond boxes, intrinsic volumes determine neither width variance nor brightness variance. The sign-pattern representation extends to bodies whose central symmetral is a zonotope, while the cellwise co-area method applies to every full-dimensional polytope.
Comments56 pages, 2 figures
