Bilipschitz Embeddings of Metric Spaces into Space Forms |
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Authors: | Urs Lang Conrad Plaut |
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Institution: | (1) Departement Mathematik, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland;(2) Department of Mathematics, University of Tennessee, Knoxville, TN, 37996-1300, U.S.A. |
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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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Keywords: | Bilipschitz embeddings metric spaces doubling property Gromov hyperbolicity |
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