Let there be $m$ points in $\mathbb R^n$. Let $D$ be the longest distance between two of these points and let $d$ be the smallest. What is the smallest possible value of $\frac{D}{d}$ for each value of $n$ and $m$, and which configurations reach it?
It is clear when $n$ is $1$ the solution is reached when the points are evenly spaced in a line.
For $n=2$ I think the answer may be the regular polygons. For higher dimensions maybe it is reached by setting all the points in a sphere and trying to spread them out as best possible?