On Delaunay Triangulations of Gromov Sets

Let $Y$ be a subset of a metric space $X\mathrm{.}$ We say that $Y$ is $\eta$-Gromov provided $Y$ is $\eta$-separated and not properly contained in any other $\eta$-separated subset of $X\mathrm{.}$ In this paper, we review a result of Chew which says that any $\eta$-Gromov subset of ${\mathbb{R}}^2$ admits a triangulation $\mathcal{T}$ whose smallest angle is at least $\pi \mathrm{/}6$ and whose edges have length between $\eta$ and $2\eta \mathrm{.}$ We then show that given any $k = 1,2,3\ldots$,thereisasubdivision$$\mathcal{T}$ _{k}$of$$\mathcal{T}$$ whose edges have length in $[\frac{\eta }{10k},\frac{2\eta }{10k}]$ and whose minimum angle is also $\pi \mathrm{/}6$. These results are used in the proof of the following theorem in [10]: For any $k\in R,v>0,$ and $D>0,$ the class of closed Riemannian $4$-manifolds with sectional curvature $\ge k,$ volume $\ge v,$ and diameter $\le D$ contains at most finitely many diffeomorphism types. Additionally, these results imply that for any $\epsilon >0$, if $\eta >0$ is sufficiently small, any $\eta$-Gromov subset of a compact Riemannian $2$-manifold admits a geodesic triangulation $\mathcal{T}$ for which all side lengths are in $\left[ \eta \left( 1-\varepsilon \right) ,2\eta \left( 1+\varepsilon \right) \right] $ and all angles are $\ge \frac{\pi }{6}-\epsilon \mathrm{.}$