International Mathematics Competition
for University Students
2024

IMC2024: Day 1, Problem 5

Problem 5. Let \(\displaystyle n>d\) be positive integers. Choose \(\displaystyle n\) independent, uniformly distributed random points \(\displaystyle x_1,\ldots,x_n\) in the unit ball \(\displaystyle B\subset \mathbb{R}^d\) centered at the origin. For a point \(\displaystyle p\in B\) denote by \(\displaystyle f(p)\) the probability that the convex hull of \(\displaystyle x_1,\ldots,x_n\) contains \(\displaystyle p\). Prove that if \(\displaystyle p,q\in B\) and the distance of \(\displaystyle p\) from the origin is smaller than the distance of \(\displaystyle q\) from the origin, then \(\displaystyle f(p)\geq f(q)\).

Fedor Petrov, St Petersburg State University

  Solution