HMIC 2018 Problem 2

Consider a finite set of points $T \in \mathbb{R}^{n}$ contained in the $n$-dimensional unit ball centered at the origin, and let $X$ be the convex hull of $T$. Prove that for all positive integers $k$ and all points $x \in X$, there exist points $t_{1}, t_{2}, \ldots, t_{k} \in T$, not necessarily distinct, such that their centroid

$$\frac{t_{1}+t_{2}+\cdots+t_{k}}{k}$$

has Euclidean distance at most $\frac{1}{\sqrt{k}}$ from $x$. (The $n$-dimensional unit ball centered at the origin is the set of points in $\mathbb{R}^{n}$ with Euclidean distance at most 1 from the origin. The convex hull of a set of points $T \in \mathbb{R}^{n}$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)