Sylvester's problem for beta-type distributions

2024/12/31 by Anna Gusakova, Gusakova, Anna, Zakhar Kabluchko +1
Mathematics · #52A22 #FOS: Mathematics #Mathematical and Theoretical Analysis #Metric Geometry (math.MG) #Primary: 60D05 #Probability (math.PR) #Secondary: 52A40 #Statistical Distribution Estimation and Applications #Statistical and numerical algorithms

paper · pdf · doi:10.48550/arxiv.2501.00671

openalex publication_date 2024/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider d+2 i.i.d. random points X₁,…, X_d+2 in \mathbb R^d. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings: (i) the distribution of X₁ is multivariate standard normal; (ii) the density of X₁ is proportional to (1-‖x‖²)^β on the unit ball (the beta distribution); (iii) the density of X₁ is proportional to (1+‖x‖²)^-β (the beta prime distribution). In the Gaussian case, we show that this probability equals twice the sum of the solid angles of a regular (d+1)-dimensional simplex.

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