EDIT: In physics, this is equivalent to the problem of finding the equilibrium configuration of $k$ electrons arranged on an $n$-sphere. This is also known as the Thomson problem.
Does anyone know of a way to find a set of vectors $(v_i)_{i=1}^k$ on the unit $n$-sphere such that they are maximally orthogonal to each other? That is, given the function for minimal angle between $k$ $n$-vectors, $f:(\mathbb{R}^n)^k\to\mathbb{R} $:
$$f(u_1, u_2, \dots, u_k)=\left|\min_{1\le i<j\le k}\angle(u_i, u_j)\right|$$ I look for a subset of the unit $n$-sphere that maximizes that function:
$$ (v_i)_{i=1}^k:=\arg \max \{f(u_1, u_2, \dots, u_k)\mid u_i\in S^n\}$$
There's obviously a degeneracy in the choice of the $u_i$'s, but that isn't an issue. For $k\le n$ the problem is simple, so assume $k>n$.
Thank you very much!