Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define $$ a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$ where $\mu$ is Lebesgue measure and $B(x,r)$ is the ball of radius $r$ centered at $x$.
Clearly $a$ is an increasing, piecewise differentiable function.
Is there an upper bound on the derivative $a'(r)$ for $r_1\le r\le r_2$, depending only on $r_1$ and $r_2$ but not on $m$ or the points $x_i$, assuming $0<r_1<r_2<\infty$?
Clearly, if all the balls are disjoint, $a'(r)$ is proportional to $mr^{n-1}$. For any given $r$ we can have $O(r^{-n})$ disjoint balls present, leading to $a'(r)=O(r^{-1})$. Hence the lower bound $r_1$ is necessary. The upper bound is necessary because $a(r)$ would be roughly proportional to $r^n$ when $r$ is very large.
I think a positive answer could lead to a positive answer to this other question, and that is why I ask.
