The diameter of a subset $X$ of $\mathbb{R}^n$ is defined as $\sup\{|x-y|:x,y\in X\}$.
What is the smallest radius $r(d,n)$ such that any subset $X$ of diameter $d$ in $\mathbb{R}^n$ is contained in a ball of radius $r(d,n)$? What are the $X$ that realize this bound? I know that $r(d,n)\leq d$. The equilateral triangle gives $r(d,2)\geq d/\sqrt(3)$ and I think we have equality here but I don't know how to prove it.
Answer: $r(d,n) = d\sqrt{\frac{n}{2(n+1)}}$.
This is known as Jung's theorem (see also this question). The extremal case is the regular $n$-simplex in each dimension.