Let $p_1, \ldots, p_k$ be $k$ points in $\mathbb{R}^n$ so that $\max_{i,j}\|p_i - p_j\| = \epsilon$ where we are employing the standard Euclidean norm.
What is the smallest $r > 0$ so that there exists some $x \in \mathbb{R}^n$ with $\|x - p_i\| \leq r$ for all $1 \leq i \leq n$?
And most importantly,
How does $r$ change with $\epsilon$, $k$ and $n$?