There are $n$ points on a plane, such that distance between any two of them $\geq 1$. Question is, what is minimum possible diameter for such set of points, that is minimum of distance between two farthest points. At first I thought optimal configuration would be regular $n$-gon with side 1. For this configuration the answer is $$\frac{\sin(\frac{\pi}{n}\cdot\lfloor{\frac{n}{2}}\rfloor)}{\sin(\frac{\pi}{n})}$$
For $n=2,3$, this is obviously true, answer is $1$. however for $n=7$, There is a configuration with diameter $2$, regular hexagon with side $1$, and its center.
I can't prove that regular polygon is answer for $n=4,5,6$. But I still think that it is optimal, So how can I prove it for $n=4,5,6$ and what is the answer in general case?