Central limit theorems for random boundary polytopes

2025/09/24 by Reitzner, Matthias, Sonnleitner, Mathias
#52A22 (Primary) 60D05 (Secondary) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.20058

Abstract

The number of faces of the convex hull of n independent and identically distributed random points chosen on the boundary of a smooth convex body in ℝ^d is investigated. In dimensions two and three the number of k-faces is known to be constant almost surely and in dimension four and higher the variance is known to be non-zero if k≥ 1. We show that it is of order n. This is complemented by a central limit theorem with a Berry-Esseen bound which is of optimal order n^-1/2. We derive similar results for the Poissonized model, where additionally the number of random points is Poisson distributed. As a main tool, we develop a representation of the number of faces as a sum of exponentially stabilizing score functions.

Citations

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Related

• Central limit theorems for Gaussian polytopes
• Limit theorems for random polytopes with vertices on convex surfaces
• Central limit theorems for random polytopes in a smooth convex set
• The convex hull of random points on the boundary of a simple polytope
• Random polytopes: central limit theorems for intrinsic volumes