Sylvester's problem for beta-type distributions

TL;DR

This paper calculates the probability that the convex hull of $d+2$ i.i.d. points in $\,\mathbb{R}^d$ forms a simplex under three specific distributional settings, including Gaussian, beta, and beta prime distributions.

Contribution

It provides explicit probability formulas for the convex hull being a simplex for three important classes of distributions in arbitrary dimensions.

Findings

Gaussian case: probability equals twice the sum of solid angles of a regular simplex.

Derived explicit formulas for beta and beta prime distributions.

Extended Sylvester's problem to new distributional settings in high dimensions.

Abstract

Consider $d+2$ i.i.d. random points $X_1,\ldots, X_{d+2}$ 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 $X_1$ is multivariate standard normal; (ii) the density of $X_1$ is proportional to $(1-\|x\|^2)^{\beta}$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+\|x\|^2)^{-\beta}$ (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.