Bilipschitz Embeddings of Metric Spaces into Space Forms
Bilipschitz Embeddings of Metric Spaces into Space Forms
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Bilipschitz 将度量空间嵌入到空间形式中
DOI：
10.1023/a:1012093209450
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发表时间：
2001
影响因子：
0.5
通讯作者：
C. Plaut
中科院分区：
文献类型：
--
作者：
U. Lang;C. Plaut
本文介绍了一些基本的几何工具来构造度量空间的bilipschitz嵌入到（有限维）欧氏或双曲空间。其中一个主要结果是：如果X是一个具有凸距离函数的测地度量空间，且具有测地段可扩张到射线的性质，则X允许bilipschitz嵌入到某个欧氏空间当且仅当X具有加倍性质，且X允许bilipschitz嵌入到某个双曲空间当且仅当X是Gromov双曲的且在某种尺度上加倍.在这两种情况下，嵌入的图像被证明是目标空间中的Lipschitz收缩，假设X是完全的。
The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.