This problem arose in some of my own personal data science research and I am wondering if anyone has encountered this before.
Consider $k$ points that lie on an $n$-dimensional unit sphere such that all pairs of points are as far apart as possible. Is there a closed-form solution to finding this distance?
First I tried solving it in a simpler case of a 2-dimensional sphere, i.e a circle. In this case, the problem reduces to finding the length of one side of a regular polygon inscribed in a unit circle. Using some simple trigonometry we can derive a formula for the length of one side of this polygon:
$$d = 2 \sin \frac{\pi}{k}$$
where $d$ is the distance between any pair of points and $k$ is the number of points.
So in 2-dimensions, this is the furthest distance between any two points on a unit circle such that all pairs of points are the same distance apart. It seems that one could similarly solve this in 3-dimensions by inscribing solids inside the sphere and finding their side lengths, though I am unsure of how to approach this, let alone $n$-dimensions.
Is it possible to find a general solution to this problem when we have $k$ points lying on an $n$-dimensional unit sphere?
Thank you to anyone who can offer insights or help! Please let me know if anything needs further clarification.