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Consider the unit sphere $S^n$ in ${\bf R}^{n+1}$.

Consider $S(r)$, a union of $r$-balls in $S^n$ which is disjoint and that $S(r)$ has maximum area.

Then define $$ c_n\doteq \lim_{r\rightarrow 0} \frac{{\rm vol}\ S(r)}{{\rm vol}\ S^n}. $$

Here I have a question : $c_n=1$ for each $n$ ?

If not, please explain about it. Thank you in advance.

HK Lee
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    What does the equality sign with a dot above means? – Marra Jul 29 '13 at 07:17
  • I mean the definition. That is, the constant $c_n$ is the limit of ${\rm vol} \ S(r)/{\rm vol}\ S^n$ as $r \rightarrow 0$. – HK Lee Jul 29 '13 at 07:19
  • Well, it seems to be true for me; why are you unsure of this? – Marra Jul 29 '13 at 07:24
  • You can try to make estimatives and then use the Squeeze Theorem, I guess. – Marra Jul 29 '13 at 07:25
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    The motivation is originated from Chapter 2 (page 33) in the book " Metric Structures for Riemannian and Non-Riemannian Spaces - Gromov ". Even though I think that $c_n=1$ at first, consider the unit square which contains $n^2$-balls of radius $\frac{1}{2n}$. Then the limit of $n^2 \pi (\frac{1}{2n})^2$ is $\frac{\pi}{4} < 1$ – HK Lee Jul 29 '13 at 07:38

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If I understand your question correctly, it seems to me that this is just the sphere-packing problem (or also the hypersphere packing problem). The fact that you're packing the spheres in a larger sphere shouldn't matter because we allow the smaller spheres to be arbitrarily small.

Here is a nice presentation on some of what we know. It seems much of what you are asking is still an open question. In summary, it seems that $c_0=1$, $c_1=\frac{\pi}{2\sqrt 3}$, and $c_2=\frac{\pi}{3\sqrt 2}$.

Some items of note: the sphere-packing problem is often discussed in two sub-problems: regular (lattice) packings and irregular packings. Also, we know more about $c_7$ and $c_{23}$ than we do about $c_4$ because of special lattices that exist in $\mathbb{R}^8$ and $\mathbb{R}^{24}$. A related problem is the kissing number problem, which again is solved in $8$ and $24$ dimensions, but open in $5$ dimensions!

Jared
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