Let $\mathbb{S}_{d-1}$ denote the unit $(d-1)$-sphere. Let \begin{align} f(d, n) = \operatorname*{argmin}_{x \in (\mathbb{S}_{d-1})^n} \sum_{i \in n} \max_{j \in n - \{i\}} x_i \cdot x_j \end{align}
Let $g(d, n)$ be the corresponding minimum value. What is known about $f$ and $g$? Do they have a name in the literature? Are the following solutions correct? Are other solution classes known?
| $d$ | $n$ | $f(d,n)$ |
|---|---|---|
| $1$ | $n$ | $\lfloor n/2 \rfloor$ on one point and $\lceil n/2 \rceil$ on the other |
| $2$ | $n$ | vertices of a regular $n$-gon |
| $d$ | $n \leq d+1$ | vertices of a regular $(n-1)$-simplex |
| $3$ | $5$ | vertices of a triangular bipyramid |
| $3$ | $6$ | vertices of a regular octahedron |
| $3$ | $8$ | vertices of a uniform square antiprism |
| $3$ | $12$ | vertices of a regular icosahedron |
See the Thomson problem and spherical codes for related problems. These questions might have relevant information:
