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Let $B$ be a closed ball in $\mathbb R^d$ of radius $R$ and let $N=N_R(r)$ denote the maximal cardinality of the $r$-separated sets (meaning any two points in this set have distance at least $r$) that can be contained in $B$.

If $r>2R$ then $N_R(r)=1$.

If $r\le 2R$, I expect $N$ can be bounded by something like $C(\frac{2R}{r})^{d+\delta}$ where $C,\delta$ are absolute constants (I may be wrong here. Let me know the correct form).

I wonder what are naive/classical/latest results on this estimation.

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    $\begingroup$ Hint: compare volumes. This is naive/classical and not so bad. You get the bound $(\frac{2R}{r}+1)^d$. $\endgroup$ – Mikael de la Salle Oct 8 at 7:39
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    $\begingroup$ The key observation is that balls of radius $r/2$ centered in the separated set cannot intersect and have to lie completely in a ball of radius $R+r/2$. Conversely any such balls of radius $r/2$ in $B_{R+r/2}$ have a center in $B_R$ and if they do not intersect their centers have distance at least $r$, so your problem is equivalent to a sphere packing problem, for which there should be a lot of advanced results. $\endgroup$ – mlk Oct 8 at 8:05

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