Expected f-vector of the Poisson Zero Polytope and Random Convex Hulls\n in the Half-Sphere

2019/01/29 by Zakhar Kabluchko, Kabluchko, Zakhar
Computer Science · Mathematics · #51M20 #52A20 #52A55 #60D05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Primary: 52A22 #Probability (math.PR) #Secondary: 52B11

paper · pdf · doi:10.48550/arxiv.1901.10528

openalex publication_date 2019/01/29 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

We prove an explicit combinatorial formula for the expected number of faces\nof the zero polytope of the homogeneous and isotropic Poisson hyperplane\ntessellation in mathbb R^d. The expected f-vector is expressed through the\ncoefficients of the polynomial (1+ (d-1)²x²) (1+(d-3)² x²) (1+(d-5)²\nx²)
ldots. Also, we compute explicitly the expected f-vector and the\nexpected volume of the spherical convex hull of n random points sampled\nuniformly and independently from the d-dimensional half-sphere. In the case\nwhen n=d+2, we compute the probability that this spherical convex hull is a\nspherical simplex, thus solving an analogue of the Sylvester four-point problem\non the half-sphere.\n

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