Critical angles in polyhedral convex cones: Numerical and statistical considerations
The maximal angle of a nontrivial closed convex cone \(K\) in Euclidean space \({\mathbb R}^n\) is defined as the nonnegative number \(\theta_{\max}(K) = \max_{u,v\in K\cap {\mathbb S}_n}\arccos\langle u, v\rangle\), where \(\langle u, v\rangle\) is the usual inner product, while \({\mathbb S}_n\) is the unit sphere in \({\mathbb R}^n\). \(K\) is nontrivial, if \(K\) is different from \(\{0\}\) and from the whole space \({\mathbb R}^n\). \((u, v)\) is a critical pair of \(K\) means that \(v-\langle u, v\rangle\,u\in K^+\) and \(u-\langle u, v\rangle\,v\in K^+\), where the symbol \(K^+\) stands for the positive dual cone of \(K\). A critical pair \((u, v)\) and the corresponding critical angle \(\theta(u, v)= \arccos\langle u, v\rangle\) are said to be proper if \(u\) and \(v\) are not collinear. The set of all proper critical angles of \(K\) is called the angular spectrum of \(K\). The number \(\theta_{\max}(K)\) is the largest critical angle of \(K\). This critical angle is proper if and only if \(K\) is pointed and not a ray. Special attention is paid to the class of polyhedral convex cones because the angular spectrum of a convex cone of the form \(K= cone\{g^1,\dots, g^p\}\) is a finite set (see [1], Theorem 8.5) [\textit{A. Iusem} and \textit{A. Seeger}, Math. Program. 120, No.~1 (B), 3--25 (2009; Zbl 1163.52003)] and, what is even important, it can be computed explicitly by solving a series of generalized eigenvalue problems involving the vectors \(g^1, \dots, g^p\). The paper is organized as follows. A necessary technical background is provided by Section 2 titled ``Preliminary results''. Throughout Section 3 titled ``Numerical experimentation'', one assumes that \(K\) is a polyhedral convex cone in \({\mathbb R}^n\) described by a generator set \(\{g^1,\dots, g^p\}\). The dimension \(n\) of the underlying Euclidean space is assumed to be at least 3. Angular spectra are computed by solving appropriate generalized eigenvalue problems. In particular, in subsection 3.2 and in Section 4 titled ``Probabilistic consideration'' the authors concentrate on random polyhedral convex cones in \({\mathbb R}^n\). Finally, In Section 5 titled ``Conclusions'' the main results of the paper are summarized.
scientific article; zbMATH DE number 5675096
scientific article