Random Width and Brightness: Polyhedral Density Theory, Reconstruction, and Gaussian Identifiability

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.