AI 中文总结
本文为三维凸体的随机宽度与亮度建立正逆向理论，推导多面体精确密度，并证明基于低阶矩的逆向重构与高斯相关矩阵可辨识性定理。
AI 中文摘要
设U均匀分布于单位球面上。我们为三维凸体的随机宽度w_K(U)和亮度b_K(U)发展了一套自洽的正向与逆向理论。对于每个全维多面体，一个全局球面余面积公式将宽度密度表示为由其差体的法扇所决定的有限个角孔径之和；特别地，该密度是分段实解析的，且具有一个有限且由几何决定的关键集。该理论给出了正四面体宽度的精确密度，解决了Finch提出的一个问题，并给出了正截角八面体的精确密度，连同四面体亮度定律和等价的菱形十二面体宽度定律。在逆向方面，当观测三角形张成相关图的圈空间时，二阶和三阶极化余弦变换矩可重构有限标记方向系统；符号图切换描述了不可避免的模糊性。相反，相等的三维内蕴体积既不决定宽度定律也不决定亮度定律，即使对于中心对称体也是如此。去除空间秩约束后，得到一个关于中心化多元折叠正态向量的无维可辨识性定理：成对绝对矩和一个锚定的三元绝对矩族（对于完全相关图，包含|m - 1|^2个标记观测）在不使用四阶矩的情况下，确定相关矩阵至对角符号共轭。进一步，一个调和分解将二阶方差贡献识别为加权框架算子无迹部分的Frobenius范数平方的常数倍，并解释了为何该贡献在不可约对称下消失。
英文摘要
Let U be uniformly distributed on the unit sphere. We develop a self-contained forward and inverse theory for the random width w_K(U) and brightness b_K(U) of three-dimensional convex bodies. For every full-dimensional polytope, a global spherical co-area formula expresses the width density as a finite sum of angular apertures determined by the normal fan of its difference body; in particular, the density is piecewise real analytic with a finite geometrically determined critical set. This theory yields exact densities for the width of the regular tetrahedron, resolving a question of Finch, and for the regular truncated octahedron, together with the tetrahedral brightness law and the equivalent rhombic-dodecahedral width law. On the inverse side, second- and third-order polarized cosine-transform moments reconstruct finite labelled direction systems whenever the observed triangles span the cycle space of the correlation graph; signed-graph switching describes the unavoidable ambiguity. In contrast, equal three-dimensional intrinsic volumes do not determine either the width law or the brightness law, even for centrally symmetric bodies. Removing the spatial rank constraint gives a dimension-free identifiability theorem for centered multivariate folded-normal vectors: pairwise absolute moments and an anchored family of triple absolute moments, comprising |m - 1|^2 labelled observations for a complete correlation graph, determine the correlation matrix up to diagonal sign conjugacy without fourth-order moments. A harmonic decomposition further identifies the degree-two variance contribution as a constant multiple of the squared Frobenius norm of the traceless part of the weighted frame operator and explains why this contribution vanishes under irreducible symmetry.
Comments92 pages, 4 figures
