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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References
[ABi] Alexander, S. B. and Bishop, R. L.: The Hadamard Cartan theorem in locally con-vex metric spaces, Enseign. Math. 36 (1990), 309-320.
[Al] Almgren, F. J.: The homotopy groups of the integral cycle groups, Topology 1 (1962), 257-299.
[AmK] Ambrosio, L. and Kirchheim, B.: Currents in metric spaces, Acta Math. 185 (2000), 1-80.
[As1] Assouad, P.: Étude d'une dimension métrique liée à la possibilitéde plongement dans ℝ”, C. R. Acad. Sci. Paris 288 (1979), 731-734.
[As2] Assouad, P.: Plongements Lipschitziens dans ℝ”, Bull. Soc. Math. France 111 (1983), 429-448.
[B] Berestovskii, V. N.: Introduction of a Riemannian structure in certain metric spaces, Siberian Math. J. 16 (1975), 210-221.
[BoS] Bonk, M. and Schramm, O.: Embeddings of Gromov hyperbolic spaces, Preprint.
[BrH] Bridson, M. R. and Haefliger, A.: Metric Spaces of Non-Positive Curvature, Springer, New York, 1999.
[C] Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 413-460.
[G1] Gromov, M.: Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1-147.
[G2] Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math. 152, Birkhäuser, 1999.
[HuW] Hurewicz, W. and Wallman, H.: Dimension Theory, Princeton Univ. Press, 1948.
[L] Laakso, T. J.: Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaréinequality, Geom. Funct. Anal. 10 (2000), 111-123.
[La] Lang, U.: Higher-dimensional linear isoperimetric inequalities in hyperbolic groups, Internat. Math. Res. Notices 2000, 709-717.
[La+] Lang, U., Pavlović, B. and Schroeder, V.: Extensions of Lipschitz maps into Hadamard spaces, Geom. Funct. Anal. 10 (2000), 1527-1553.
[Lu] Luosto, K.: Ultrametric spaces bi-Lipschitz embeddable in ℝ”, Fundam. Math. 150 (1996), 25-42.
[Luu] Luukainen, J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1998), 23-76.
[LuuMo] Luukainen, J. and Movahedi-Lankarani, H.: Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fundam. Math. 144 (1994), 181-193.
[M] McShane, E. J.: Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[P] Pansu, P.: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math. 129 (1989), 1-60.
[Pa] Pauls, S. D.: The large scale geometry of nilpotent Lie groups, Preprint.
[Se1] Semmes, S.: Bi-Lipschitz mappings and strong A∞ weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), 211-248.
[Se2] Semmes, S.: On the nonexistence of bilipschitz parameterizations and geometric problems about A ∞-weights, Rev. Mat. Iberoamer. 12 (1996), 337-410.
[Se3] Semmes, S.: Metric spaces and mappings seen at many scales, Appendix B in [G2].
[Se4] Semmes, S.: Bilipschitz embeddings of metric spaces into Euclidean spaces, Publ. Mat. 43 (1999), 571-653.
[St] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
[We] Wenger, S.: Isoperimetric inequalities in weakly convex metric spaces, diploma thesis, ETH Zürich, March, 2000.
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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