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

For a $d$-dimensional random vector $X$, let $p_{n, X}(θ)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $θ$. We provide several sharp inequalities regarding $p_{n, X}(θ)$ and $N_X(θ)$ denoting the smallest $n$ for which $p_{n, X}(θ)\ge1/2$. As a main result, we derive the totally general inequality $1/2 \le α_X(θ)N_X(θ)\le 3d + 1$, where $α_X(θ)$ (a.k.a. the Tukey depth) is the minimum probability that $X$ is in a fixed closed halfspace containing the point $θ$. 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.

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