Omri Abas | Searcharxiv

arXiv2026

Width Laws and Spectral Geometry

We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a $d$-dimensional orthotope, we prove an exact parity law for the maximal $π^{-1}$-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first $d$ scalar width moments recover the unordered side vector, and $d-1$ moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies $M_{n,a} = λ_n(a)^{1/2}/π$. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.

math.MG↗

arXiv2026

Width distributions for rectangular boxes

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.

math.MG↗