| Popis: |
Let K be a convex set in R d and let K λ be the convex hull of a homogeneous Poisson point process P λ of intensity λ on K. When K is a simple polytope, we establish scaling limits as λ → ∞ for the boundary of K λ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K λ , k ∈ { 0 , 1 , . . . , d − 1 } , resolving an open question posed in [17] . The scaling limit of the boundary of K λ and the variance asymptotics are described in terms of a germ–grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d − 1 × R having intensity d e d h d h d v . |
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