Abstract
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.
Highlights
The following is a literal quotation from [7]: “Recent work has exposed a phenomenon of abrupt phase transitions in high-dimensional geometry
We quote from the work of Amelunxen et al [1], which discovers and describes several of these phenomena: “This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems . . . The applied results depend on foundational research in conic geometry.”
For n ∈ N, the (φ, n)-Cover–Efron cone Cn is defined as the positive hull of n independent random vectors X1, . . . , Xn with distribution φ, under the condition that this positive hull is different from Rd
Summary
The following is a literal quotation from [7]: “Recent work has exposed a phenomenon of abrupt phase transitions in high-dimensional geometry. For n ∈ N, the (φ, n)-Cover–Efron cone Cn is defined as the positive hull of n independent random vectors X1, . This may seem unexpected, since the random cones considered in [9] and here have different distributions (see, the appendix). Cn = Sn◦ in distribution, where Sn◦ denotes the polar cone of Sn. For the expectations appearing in our theorems, explicit representations are available.
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