Estimating the probability that a given vector is in the convex hull of a random sample

TL;DR

This paper establishes sharp inequalities for the probability that a point lies in the convex hull of random vectors, with applications to randomized cubature, measure reduction, and convex body inclusion.

Contribution

It derives a general inequality relating Tukey depth, sample size, and dimension, and applies it to bounds on convex hull probabilities and random convex bodies.

Findings

Derived the inequality 1/2 ≤ α_X(θ)N_X(θ) ≤ 3d + 1.

Provided a moment-based bound on N_X( E[X] ).

Generalized results in random matrix theory regarding convex bodies.

Abstract

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and $N_X(\theta)$ denoting the smallest $n$ for which $p_{n, X}(\theta)\ge1/2$. As a main result, we derive the totally general inequality $1/2 \le \alpha_X(\theta)N_X(\theta)\le 3d + 1$, where $\alpha_X(\theta)$ (a.k.a. the Tukey depth) is the minimum probability that $X$ is in a fixed closed halfspace containing the point $\theta$. We also show several applications of our general results: one is a moment-based bound on $N_X(\mathbb{E}[X])$, which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of $X$, where our combinatorial approach allows us to generalize existing results in random matrix community significantly.