Consider $n$ unit vectors $v_i \in \mathbb{R}^d$ in the $d$-dimensional Euclidean space. Assume that the angle between any pair of two vectors $v_i$ and $v_j$ is the same. Then, what would be the maximum for this angle?
I think the solution is quite easy for the $d>n$ case since in this case the vectors form a simplex.
I am curious for the solutions in the general case of $n>d$.
I found some posts ([link1][1][link2][2][link3][3]) that discuss a similar problem, but couldn't really find a solution to this.
Also, the [paper][4] here seems to calculate the estimated value for the maximum angle; namely,Eq. (7) in the appendix of the paper $$ A \approx n^{-\frac{2}{d-1}} \Gamma(1 + \frac{1}{d-1}) \left( \frac{\Gamma(\frac{d}{2})}{2 \sqrt{\pi} (d-1) \Gamma(\frac{d-1}{2})} \right)^{-\frac{1}{d-1}}. $$ I am wondering if this estimation is actually correct (the paper lacks the derivation of this approximation).
Using Wolfram, I computed the value for $n=10^4$ with $d=512$, obtaining $A \approx 0.94$. Would this make sense?
[1]: https://math.stackexchange.com/questions/3778459/maximum-angle-of-separation-between-n-vectors-in-m-dimensions#:~:text=In%202D%20Given%202,simply%20divide%20360%20by%20n). [2]: How to determine maximum angles between vectors? [3]: Is this really an open problem? Maximizing angle between $n$ vectors [4]: https://arxiv.org/abs/1801.07698
