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

Understand

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 $\theta$.