Threshold Phenomena for Random Cones

We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index d−k when also k tends to infinity.