On Delaunay Triangulations of Gromov Sets

TL;DR

This paper reviews a triangulation result for $ ext{eta}$-Gromov subsets in $ ext{R}^2$, introduces refined subdivisions with controlled edge lengths and angles, and applies these to finiteness theorems for Riemannian 4-manifolds.

Contribution

It extends Chew's triangulation result to finer subdivisions with uniform edge lengths and angles, and applies these to geometric finiteness results in Riemannian geometry.

Findings

Existence of subdivisions with smaller edge lengths maintaining angle bounds

Application to finiteness of diffeomorphism types of Riemannian 4-manifolds

Approximate geodesic triangulations with controlled side lengths and angles

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 $\geq \frac{\pi }{6}-\varepsilon .$