论文标题
关于Gromov套装的Delaunay三角
On Delaunay Triangulations of Gromov Sets
论文作者
论文摘要
让$ y $是公制空间$ x的子集。三角剖分$ \ MATHCAL {T} $的最小角度至少为$π/6 $，其边缘的长度在$η$和$2η之间。 \fracη{10 k}，\ frac {2η} {10 k} \ right] $，其最小角度也为$π/6 $。 这些结果用于[10]中的以下定理证明：对于任何$ k \ in r，v> 0，$和$ d> 0，$ n of the of liemannian $ 4 $ - manifolds，带有截面曲率$ \ geq k，$ polume $ \ polume $ \ geq v，$ \ geq v，$ and $和diameter $ \ leq d $包含最适合的diffefim，此外，这些结果暗示，对于任何$ \ varepsilon> 0 $，如果$η> 0 $足够小，则任何$η$ - gromov子集的紧凑型Riemannian $ 2 $ 2 $ -MANIFOLD承认一个大地测量的三角形三角形$ \ Mathcal {t} $ for $ \ ev ph ev ev ev ev左来[ ，2η\ left（1+ \ varepsilon \ right）\ right] $，所有角度均为$ \ geq \ frac {π} {6} {6} - \ varepsilon。
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 $\geq \frac{π}{6}-\varepsilon .$
