Question 11
Consider a finite set of points $T \subset \mathbb{R}^n$ 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 $T \subset \mathbb{R}^n$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)
$$
\frac{t_1 + t_2 + \cdots + t_k}{k}
$$
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 $T \subset \mathbb{R}^n$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)