What's the maximum number of points at distance r or more inside a d-dimensional sphere of radius r?

Question

We have a sphere of radius $r$ in a $d$-dimensional space. What is the maximum amount of points that I can fit inside the sphere such as the distance between any pair of points is at least $r$? And strictly bigger than $r$?

I believe this is equivalent to packing d-dimensional spheres of radius r/2 inside a sphere of radius r.

If you have an idea on the order of the answer I would also appreciate it.

Thanks!

Related questions:

• This question says that the number is 12 for d=3, what about for a general d?

• As opposed to this question, I'm only concerned for points at distance $r$, not any arbitrary distance.

Answer 1

In general, I believe you are seeking the kissing number. In $d=2$, $6$ circles can touch a central $7$th, so $7$ points at pairwise distance $\ge r$ can be packed:

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In $d=3$, the kissing number is $12$, so you could surround one sphere with $12$, leading to $13$ (not $12$ as you say) points in the sphere each pair separated by at least $r$.

In dimension $d=4$, you would place the sphere centers at the vertices of the $24$-cell, and a $25$-th sphere in the center.

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In $d=24$, you could pack $196560+1$ points.