Search

Let \mathcal Z_d be the zero cell of a d-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of k-dimensional faces of \mathcal Z_d, as d\to \infty. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to \sqrt {2\pi /3}\, d^{3/2}, as d\to \infty. We also prove that the expected solid angle of a random cone spanned by d random vectors that are independent and uniformly distributed on the unit upper half-sphere in \mathbb R^{d} is asymptotic to \sqrt 3 \pi ^{-d}, as d\to \infty. [ABSTRACT FROM AUTHOR]

We consider d-dimensional random vectors Y1, . . . ,Yn that satisfy a mild general position assumption a.s. The hyperplanes (Yi−Yj)⊥ (1 ≤ i < j ≤ n) generate a conical tessellation of the Euclidean d-space which is closely related to the Weyl chambers of type An−1. We determine the number of cones in this tessellation and show that it is a.s. constant. For a random cone chosen uniformly at random from this random tessellation, we compute expectations of several geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conical quermassintegrals. Under the additional assumption of exchangeability on Y1, . . . ,Yn, the same is done for the dual random cones which have the same distribution as the positive hull of Y1−Y2, . . . , Yn−1−Yn conditioned on the event that this positive hull is not equal to Rd. All these expectations turn out to be distribution-free. Similarly, we consider the conical tessellation induced by the hyperplanes (Yi + Yj)⊥ (1 ≤ i < j ≤ n), (Yi − Yj)⊥ (1 ≤ i < j ≤ n), Yi⊥ (1 ≤ i ≤ n). This tessellation is closely related to the Weyl chambers of type Bn. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected non-trivially by a certain linear subspace in general position. [ABSTRACT FROM AUTHOR]