  Let $G$ be a graph which satisfies $c^{-1} a^r \le |B(v,r)| \le c a^r$, for
some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that
this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+
|A|)$ for some constant $C=C(a,c)$ and every finite set of vertices $A$.
