IMC 2024 · Problem 5

Let $n > d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1, \ldots, x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1, \ldots, x_n$ contains $p$. Prove that if $p, q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$.