You are given a circle of radius $1$. Suppose you pick $n$ independent points randomly on the circle and join neighboring points with lines to create chords. What is the expected length of the shortest chord?
I thought about doing this using angles from the center, i.e. minimize $E[\min 2\sin(\theta_i / 2)]$ where $\theta_i$ are random on $[0, 2\pi]$ and sum of $\theta_i = 2\pi$. But not really sure where to go about it from here.
I think you might find the CDF for the minimum chord length might be something like $$\displaystyle \Pr(C_{\min} \le c) = 1-{\left( 1-\frac{n}{\pi} \sin^{-1}\left( \frac{c}{2} \right) \right) }^{n-1}$$
so the expected minimum chord length might be something like $$\displaystyle \int_{x=0}^{2\sin(\pi/n)}{\left( 1-\frac{n}{\pi} \sin^{-1}\left( \frac{x}{2} \right) \right) }^{n-1}\; dx.$$