You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another city, and the roads need not be straight). What is the shortest length of road needed to do this?
After playing around with it, it turns out that two diagonals is not optimal. If you assume that the road system must have the following shape
with angle $\alpha$ at all 4 corners, basic calculus shows that $\alpha=\frac\pi6$ minimises road usage in this form, using $(1+\sqrt3)$ km of road. The book I found this problem in also indicates this is the optimal shape to minimise the amount of road. However, I now have the following questions:
How can one prove or disprove that the system above with $\alpha=\frac\pi6$ is optimal for minimising roads? If it isn't, what system is?
Is there a way to figure out how to minimise road usage for cities on the vertices of any regular $n$-gon?