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

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 )ge 1/2. 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}!left[ Xright] ), 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.