Random Polyhedral Cones: Distributional Results via Gale Duality
Zakhar Kabluchko
TL;DR
The work analyzes distributional properties of random polyhedral cones generated by i.i.d. points on the sphere, using linear Gale duality to relate primal cones to Gale dual configurations and to couple Gaussian projections with unit-sphere vectors. It proves a striking symmetry m(d,k) = m(k,d) for the moments of the cone's solid angle, derives explicit variance and third-moment formulas, and furnishes spherical Sylvester-type results for small n with a detailed vertex-count distribution in a key regime. A central contribution is a high-dimensional limit theorem for the number of (d−q)-faces in the regime n = d+k, where the limit is non-Gaussian and expressed as a weighted sum of independent chi-squared variables governed by eigenvalues of a convolution operator on the sphere. The results rely on a unifying Gale-duality methodology, enabling a robust analysis via U-statistics and conic intrinsic volumes, and yield both exact and asymptotic descriptions, including independence phenomena for face events and exponential concentration bounds.
Abstract
Let $U_1,\ldots,U_n$ be independent random vectors uniformly distributed on the unit sphere $\mathbb S^{d-1}\subseteq\mathbb R^d$, where $n\ge d$, and consider the random polyhedral cone \[ \mathcal W_{n,d}:=\mathop{\mathrm{pos}} (U_1,\ldots,U_n) = \{λ_1 U_1+ \ldots + λ_n U_n: λ_1\geq 0, \ldots, λ_n \geq 0\}. \] We establish several distributional results for $\mathcal W_{n,d}$ and the associated spherical polytope $\mathcal W_{n,d}\cap\mathbb S^{d-1}$. Our main contributions include: (i) Let $α_d$ denote the solid angle of $\mathcal W_{d,d}$ and write $m(d,k):=\mathbb E[α_d^k]$ for its $k$-th moment. We prove the symmetry $m(d,k)=m(k,d)$. As an application, we compute $\mathop{\mathrm{Var}}[α_d]=2^{-d}(d+1)^{-1}-4^{-d}$ and derive a closed formula for the third moment. (ii) For $n=d+1,d+2,d+3$ we determine the probability that $\mathcal W_{n,d}\cap\mathbb S^{d-1}$ is a spherical simplex, a spherical analogue of the classical Sylvester problem. In the case $n=d+2$ we also determine the distribution of the number of vertices of $\mathcal W_{d+2,d}\cap\mathbb S^{d-1}$. (iii) Let $f_\ell(\mathcal W_{n,d})$ denote the number of $\ell$-dimensional faces of $\mathcal W_{n,d}$. We prove a distributional limit theorem for $f_\ell(\mathcal W_{n,d})$ in the regime $n=d+k$ and $\ell=d-q$, where $k,q\in\mathbb N$ are fixed and $d\to\infty$. The limit law is a weighted sum of independent chi squared variables, with weights given by explicit eigenvalues of a convolution operator on the sphere. A unifying ingredient is an explicit coupling producing i.i.d. uniform vectors $U_1,\ldots,U_n\in\mathbb S^{d-1}$ together with i.i.d. uniform vectors $V_1,\ldots,V_n\in\mathbb S^{n-d-1}$ whose associated oriented matroids are Gale dual.