There exists $C=C(n,D)$ with the following property. Let $(M^{n},g)$ denote a Riemannian manifold and $U\subseteq M$ a subset such that $r_{h}(x)>r>0$ for all $x\in U$ and such that ${\rm diam}(U)\leq D\cdot r$. Then there exists an open set $U^{\prime}$ with $T_{r/2}(U)\subseteq U^{\prime}\subseteq T_{r}(U)$, such that $U^{\prime}$ has at most one of $C$ diffeomorphism types.