Abstract
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.
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Lang, U., Plaut, C. Bilipschitz Embeddings of Metric Spaces into Space Forms. Geometriae Dedicata 87, 285–307 (2001). https://doi.org/10.1023/A:1012093209450
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DOI: https://doi.org/10.1023/A:1012093209450