Face numbers of high-dimensional Poisson zero cells

2021/10/15 by Zakhar Kabluchko, Kabluchko, Zakhar · 1 citation
Mathematics · #33E20 #52A23 #52B05 #52B11 #60G55 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Point processes and geometric inequalities #Primary: 60D05 #Probability (math.PR) #Secondary: 52A22

paper · pdf · doi:10.48550/arxiv.2110.08201

openalex publication_date 2021/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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→∞. For example, we show that the expected number of hyperfaces of \mathcal Z_d is asymptotically equivalent to √(2π/3) d^3/2, as d→∞. 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 √ 3 π^-d, as d→∞.

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• On the shape of the typical Poisson-Voronoi cell in high dimensions

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