Let $p$ be a point in a space $X_{\infty}$ which is a Gromov-Hausdorff limit of manifolds in ${\mathcal{K}}(n,c,V)$. There are real numbers $b(p),r(p)>0$ and an integer $k(p)$ with the following effect. Suppose $X_{i}$ in ${\mathcal{K}}(n,C,V)$ has Gromov-Hausdorff limit $X_{\infty}$. Then there is some $k\leq k(p)$ such that for sufficiently large $i$, if $x$ is a point in $X_{i}$ with $d(x,p)\leq r(p)$ then $\rho_{k,X}(x)\geq b(p)$.