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On Delaunay Triangulations of Gromov Sets

Let $Y$ be a subset of a metric space $X.$ We say that $Y$ is $η$-Gromov provided $Y$ is $η$-separated and not properly contained in any other $η$-separated subset of $X.$ In this paper, we review a result of Chew which says that any $η$-Gromov subset of $\mathbb{R}^{2} $ admits a triangulation $\mathcal{T}$ whose smallest angle is at least $π/6 $ and whose edges have length between $η$ and $2η.$ We then show that given any $k = 1,2,3\ldots$, there is a subdivision $\mathcal{T} _{k}$ of $\mathcal{T}$ whose edges have length in $\left[ \fracη{10 k},\frac{2η}{10 k} \right] $ and whose minimum angle is also $π/6$. 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 $\geq k,$ volume $\geq v,$ and diameter $\leq D$ contains at most finitely many diffeomorphism types. Additionally, these results imply that for any $\varepsilon >0$, if $η>0$ is sufficiently small, any $η$-Gromov subset of a compact Riemannian $2$-manifold admits a geodesic triangulation $\mathcal{T}$ for which all side lengths are in $\left[ η\left( 1-\varepsilon \right) ,2η\left( 1+\varepsilon \right) \right] $ and all angles are $\geq \frac{π}{6}-\varepsilon .$