On Delaunay Triangulations of Gromov Sets

Abstract

Let $Y$ be a subset of a metric space $X.$ We say that $Y$ is $\eta $-Gromov provided $Y$ is $\eta $-separated and not properly contained in any other $\eta $-separated subset of $X.$ 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 /6 $ and whose edges have length between $\eta $ and $2\eta .$ 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[ \frac{\eta}{10 k},\frac{2\eta}{10 k} \right] $ and whose minimum angle is also $\pi /6$. 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 $\eta >0$ 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 $\geq \frac{\pi }{6}-\varepsilon .$