I came across this problem the other day and couldn't figure out a way to solve it easily or any clear answer when searching in literature.
Taking a sphere of radius $r$ in a 3 dimensional Euclidian space and taking an arbitrary number $ k \in \mathbb{N}$ of identical hard (non overlapping) spheres, is there an existing known value for their maximum radius as a function of $k$ and $r$ such that they all fit inside the first sphere?
I am here letting aside any consideration of optimal configuration and just looking for a general answer which remains true in average with random configurations. However if you only had an answer taking into account such considerations, I'd be happy any way! This problem is really close to the Sphere packing in sphere problem or to the kissing number considerations but even knowing that I wasn't able to find a clear answer, since I don't have access to the main reference of the wikipedia page (Les Cahiers du LMPA J. Liouville, 2003).
If you have any idea of the general form of the answer or any litterature advice on the subject I would really be grateful!
Thanks.