Let $X_1, \dots, X_N$ be $N$ balls of radius $R<<1$ in $[0,1]^d$ such that $N R^d \leqslant R^{\alpha}$ for some $\alpha > 0$ and $d(x_i,x_j)\geqslant 2R$ for any $i\not=j$.
The assumption means that there are very few balls and I would like to know if one can separate them : finding $\delta_1<\delta_2$ (depending only on $\alpha$ and $d$) such that for all $x_i, x_j$, $d(x_i,x_j)\leqslant R^{1-\delta_1}$ or $d(x_i,x_j)\geqslant R^{1-\delta_2}$.
Obiously, this is false in general (consider balls of radius $R$ around $\{(x,0) , \;x\in [0,1]\}$ in the two-torus) but is this is the only example ? Is it true that one can always find a partition of the balls such that two partition are distant and, in a given partition, the balls are "in the same direction" ?
