Consider the following problem, which is a variation of the sphere packing problem and is somehow related to the kissing number problem. For a dimension $n\ge 2$ and a natural $k$, let $r=r(n,k)$ be the maximal radius of $k$ spheres (in $\mathbb{R}^n$) that can be packed into the boundary ring (is that the name of it?) of radius $2r$ of the unit sphere; that is, into the set of all points whose distance from 0 is in the interval $[1-2r, 1]$.
It is easier to explain this using examples:
- $r(2,1)$ is 1, as the maximal radius $r$ of a single sphere that should be embedded in the subset of the unit sphere containing all points whose distance from 0 lies in the interval $[1-2r,1]$ is simply 1 (in which case, the said subset is the unit sphere itself). In fact, $r(n,1)=1$ for any $n\ge 2$.
- $r(2,2)$ is 0.5, as one can pack 2 0.5-radius spheres inside the unit sphere. I guess $r(n,2)=0.5$ for any $n\ge 2$.
- $r(2,3)$ is ... well, I already don't know that.
What I do know (know might be too harsh here) is that: $$\lim_{k->\infty}k\cdot r(2,k)=\pi$$
And that makes me guess $k\cdot r(2,k)$ is an increasing function, going from 1 to $\pi$. A more general limit can be described as follows: the limit of ratio of the sum of all surface areas of all packed spheres, and the surface area of the unit sphere. If we denote that ratio with $R(n,k)$, we obtain (I think) $$\lim_{k\to\infty}R(2,k)=\pi$$ and $$\lim_{k\to\infty}R(3,k)=\pi$$ but this is where it stops, as $$\lim_{k\to\infty}R(4,k)=\frac{\pi^2}{4}$$ and I think that $$\lim_{k\to\infty}R(9,k)=\frac{\pi^4}{840}$$
So my question actually consists of three sub-questions:
- Can you imagine how $r(n,k)$ (or $R(n,k)$) looks like? Is it monotone, does it always have a limit for $k\to\infty$, can you estimate that limit with respect to $n$, etc.
- I there any clear relation between $k\cdot r(n,k)$ and $R(n,k)$?
- Is $$\lim_{n,k\to\infty}R(n,k) = 0,$$ and if so, do you have any intuition about it?