Random Polyhedral Cones I: Distributional Results via Gale Duality

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