Critical angles in random polyhedral cones (Q601294)

This paper concerns the angular structure of random polyhedral cones generated by \(p\) stochastically independent vectors uniformly distributed on the unit sphere of \(\mathbb R^n\). Let \(u,v\) be unit vectors in \(K\in\Xi(\mathbb R^n)\) (here \(\Xi(\mathbb R^n)\) denotes nontrivial closed convex cones in \(\mathbb R^n\)) and \(\langle\cdot,\cdot\rangle\) is a standard inner product. One says that \((u,v)\) is a critical pair of \(K\) if \(v-\langle u,v\rangle u\in K^+\) and \(u-\langle u,v\rangle v\in K^+\) with \(K^+\) standing for the positive dual cone of \(K\). The angle \(\Theta(u,v)=\arccos\langle u,v\rangle\) formed by a critical pair is called a critical angle. The authors comment on the expected number of critical angles and the mathematical expectation of extremal angles. Finally they formulate some open problems.