This question is similar to https://en.wikipedia.org/wiki/Circle_packing_in_a_circle except I am looking for the smallest diameter, i.e: I want the smallest maximum distance between the centers of the circles, rather than the smallest circle which contains them. Asymptotically these solutions should roughly be the same of course.
This is not the same as just circle packings. For instance; two equilateral triangles of length $2m$ have a smaller area than a square of length $2m$. But the diameter of those equilateral triangles is $\sqrt3$, whereas the diameter of the square is $\sqrt2$.
I don't expect that general solutions for $n$ exist, but examples of values for small $n$ would be helpful
To be clear, by diameter I mean the maximum distance between two points.
You can solve the problem via nonlinear programming as follows. For $i\in\{1,\dots,n\}$, let $(x_i,y_i)$ be the coordinates of point $i$. The problem is to minimize $z$ subject to $$z \ge \sqrt{(x_i-x_j)^2+(y_i-y_j)^2} \ge 2$$ for $1\le i<j \le n$. Here are upper bounds (not necessarily optimal) for small $n$: \begin{array}{cc} n &z &\text{description}\\ \hline 1 &0 &\text{trivial}\\ 2 &2 &\text{trivial}\\ 3 &2 &\text{equilateral triangle}\\ 4 &2\sqrt{2}=2.8284 &\text{square}\\ 5 &1+\sqrt{5}=3.2361 &\text{regular pentagon}\\ 6 &\sqrt{2(5+\sqrt{5})}=3.8042 &\text{regular pentagon plus center}\\ 7 &4 &\text{regular heptagon}\\ 8 &\sec(3\pi/7)=4.4940 &\text{regular heptagon plus center}\\ 9 &5.1385 \\ 10 &5.5546 \\ 11 &5.7349 \\ 12 &5.8186 \\ 13 &6.3024 \\ 14 &6.6317 \\ 15 &6.9393 \\ 16 &7.1803 \\ 17 &7.5075 \\ 18 &7.6747 \\ 19 &4\sqrt{2+\sqrt3}=7.7274 \\ 20 &8.1816 \end{array} Here's $n=19$: