Suppose we have a metric space $(X,d)$ and let $a>1$. Also, let $x\in X$ and $r>0$ small enough.
Is there an upper bound $N(a)> 0$ (independent of $x,r$) so that there can fit at most $N(a)$ disjoint balls of radius $r/a$ inside the ball $B(x,r)$?
I can see that this happens if $X=\mathbb R^n$… but I use volume of balls to check this. What can we say in the general case? What about more geometric $X$? I see that this is also true, for small balls, if $X$ is a manifold.
Thanks!
Best Answer
No. E.g., consider the space $X$ comprising the coordinate axes in the space $\Bbb{R}^* = \bigcup_{n=1}^\infty\Bbb{R}^n$. $X$ looks like countably many copies of the real line identified at the origin metrized such that each pair of lines has the same metric as the union of the coordinate axes in $\Bbb{R}^2$. The open unit ball centred at the origin contains infinitely many pairwise disjoint open balls of radius $\frac{1}{2}$ centred at the points at distance $\frac{1}{2}$ from the origin. Likewise the closed unit ball centred at the origin contains infinitely many pairwise disjoint closed balls of radius $\frac{1}{3}$.