[论文解读] Random Polyhedral Cones I: Distributional Results via Gale Duality

Zakhar Kabluchko|arXiv (Cornell University)|Feb 9, 2026

Point processes and geometric inequalities被引用 0

一句话总结

论文通过 Gale 对偶性建立随机多面锥的分布结果，来自球面 IID 点的锥面分布、分量对称、高维极限定律以及面事件的独立性。

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 $d o\infty$. 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.

研究动机与目标

• Motivate the study of random polyhedral cones generated by IID uniform on the sphere and understand more than expectations (distributional results).
• Develop and exploit linear Gale duality to connect cone faces with Gale dual configurations.
• Obtain explicit distributional results for solid angles, spherical Sylvester-type questions, and high-dimensional face counts.
• Provide a coupling framework that yields independence results for face events and a U-statistic representation of face counts.

提出的方法

• Define the random cone W_{n,d} = pos(U_1, ..., U_n) with U_i IID uniform on S^{d-1}.
• Use linear Gale duality to translate cone-face questions into dual configurations and apply a unified coupling approach.
• Establish symmetry m(d,k)=m(k,d) for moments of the solid angle of W_{d,d} and compute Var(alpha(W_{d,d})).
• Derive spherical Sylvester-type probabilities for n = d+1, d+2, d+3 and, for n = d+2, the vertex distribution of W_{d+2,d} ∩ S^{d-1}.
• Prove a limit theorem for f_ell(W_{n,d}) in the regime n = d+k, ell = d-q as d → ∞, with a non-Gaussian limit involving a convolution operator on the sphere.
• Demonstrate independence of certain face events via a Gale duality coupling and provide a U-statistic representation of f_ell.

实验结果

研究问题

• RQ1 What are the distributional properties (beyond expectations) of geometric functionals of random cones generated by IID uniform sphere points?
• RQ2 Can Gale duality be leveraged to unify and transfer face-structure results between primal cones and dual configurations?
• RQ3 What are the exact moments and variances of the solid angle of W_{d,d}, and do higher moments satisfy symmetry m(d,k)=m(k,d)?
• RQ4 What are the spherical Sylvester-type probabilities for small augmentations n = d+1, d+2, d+3, and the vertex distribution when n = d+2?
• RQ5 What is the high-dimensional limit distribution of f_ell(W_{d+k,d}) as d → ∞ for fixed k,q, and is it Gaussian?

主要发现

• The paper proves symmetry m(d,k)=m(k,d) for the k-th moment of the solid angle of W_{d,d}.
• It derives Var(alpha(W_{d,d})) = 2^{-d}(d+1)^{-1} - 4^{-d} and provides a closed form for the third moment.
• For n = d+1, d+2, d+3, it computes the probability that W_{n,d} ∩ S^{d-1} is a spherical simplex; for n = d+2 it determines the vertex distribution of W_{d+2,d} ∩ S^{d-1}.
• In the high-dimensional regime n = d+k and ell = d-q (固定的 k,q), f_{d-q}(W_{d+k,d}) 标准化后收敛到非高斯极限，该极限通过球面卷积算子特征结构得到的加权卡方变量和表示.
• The paper proves independence of certain face events under specified index conditions, using a Gale duality coupling to relate face events to dual configurations.
• A U-statistic representation for f_ell(W_{n,d}) is obtained through the explicit Gale-dual coupling.