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

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