Pick each of $n$ angles , $\theta_1$ through $\theta_n$ , uniformly randomly in the range $[0,2\pi$]. Define the distance $d_{i,j}$ between $\theta_i$ and $\theta_j$ by $d_{i,j} = \min(|\theta_j - \theta_i|, 2\pi - |\theta_j - \theta_i|)$.
What is the probability that the maximal distance between every pair of angles is at most $D :D\in [0,\pi]$?
Note that the sample space $\Omega = [0,2\pi]^n$ is a hypercube of measure $(2\pi)^n$. The subspace consisting of favourable outcomes is the set of hyperplanes $H_1 \cup H_2$ where
$H_1 = \{\theta_i, \theta_j : |\theta_j - \theta_i| \leq D \}$
$H_2 = \{ \theta_i, \theta_j: 2\pi - |\theta_j - \theta_i| \leq D \}$, for all $1\leq i < j \leq n $
I can't see any simple way to compute the measure (volume) of this favourable subspace for general $n$ and want your help.