I am interested in finding a volume given by variables $\theta_1, \theta_2, \theta_3, \theta_4 \in [0, 2\pi]$ for which we have a region defined by $$ -\frac{10}{9} \geq \cos(\theta_1 - \theta_2) + \cos(\theta_1 - \theta_3) + \cos(\theta_1 - \theta_4) + \cos(\theta_2 - \theta_3) + \cos(\theta_2 - \theta_4) + \cos(\theta_3 - \theta_4) \:.$$
I posted a similar question previously here for n=3, this question is for case n=4. I am also interested if there is a more general approach to this problem as the dimensionality increases.
I will calculate more precision numerical integration so far I have
NIntegrate[
Boole[0 >=
Cos[x1 - x2] + Cos[x1 - x3] + Cos[x1 - x4] + Cos[x2 - x3] +
Cos[x2 - x4] + Cos[x3 - x4]], {x1, 0, 2*Pi}, {x2, 0, 2*Pi}, {x3,
0, 2*Pi}, {x4, 0, 2*Pi}, Method -> "LocalAdaptive"]
which gives me 957.74.
