Let $k\in\mathbb N^+$ be a parameter.
I want to find a set $S$ of $k$ points on the unit sphere in $d$ dimensions, such that the maximal distance from any point on the sphere to $S$ is minimized.
How should we pick $S$?
Can we index $S$ so that given a point $x$ on the unit sphere we can find its closest neighbor in $S$ efficiently?
For example, in two dimensions, it is easy to see that we should partition the $[0,2\pi]$ interval for the angle into $k$; e.g., for $k=3$, we can have $S=\{(1,0), (-1/\sqrt2,1/\sqrt2),(-1/\sqrt2,-1/\sqrt 2)\}$.
What should we do in higher dimensions?