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oai:www.ideals.illinois.edu:2142/24329 2023-07-10 col_2142_16340 col_2142_5131 com_2142_16339 com_2142_8903 com_2142_5130

Tyson, Jeremy T. Wu, Jang-Mei Tyson, Jeremy T. D'Angelo, John P. Merenkov, Sergiy A. Seo, Jeehyeon 2011-05-25T15:02:38Z 2011-05-25T15:02:38Z 2011-05-25T15:02:38Z 2011-05 We characterize uniformly perfect, complete, doubling metric spaces which embed bi-Lipschitzly into Euclidean space. Our result applies in particular to spaces of Grushin type equipped with Carnot-Carath ́eodory distance. Hence we obtain the first example of a sub-Riemannian manifold admitting such a bi-Lipschitz embedding. Our techniques involve a passage from local to global information, building on work of Christ and McShane. A new feature of our proof is the verification of the co-Lipschitz condition. This verification splits into a large scale case and a local case. These cases are distinguished by a relative distance map which is associated to a Whitey-type decomposition of an open subset Ω of the space. We prove that if the Whitney cubes embed uniformly bi-Lipschitzly into a fixed Euclidean space, and if the complement of Ω also embeds, then so does the full space. Item withdrawn by Alexis Thompson (athmpsn1@illinois.edu) on 2011-04-18T17:51:05Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 5 SEO_JEEHYEON.tex: 114824 bytes, checksum: 33639eed0e7fc153244249217d0bd826 (MD5) thesis.bib: 26042 bytes, checksum: 04ba256790542e6e9374018b51334170 (MD5) uiucthesis10.cls: 17916 bytes, checksum: d64797e481cef86b601a58fa5dab7ade (MD5) SEO_JEEHYEON.bib: 26024 bytes, checksum: 96b66eb030dd85236964a00e228ac3f0 (MD5) SEO_JEEHYEON.pdf: 529925 bytes, checksum: f341d235a0579b5336fd5dfe906c2bf5 (MD5) Made available in DSpace on 2011-05-25T15:02:38Z (GMT). No. of bitstreams: 6 uiucthesis10.cls: 17916 bytes, checksum: d64797e481cef86b601a58fa5dab7ade (MD5) thesis.bib: 26042 bytes, checksum: 04ba256790542e6e9374018b51334170 (MD5) Seo_Jeehyeon.pdf: 529925 bytes, checksum: 56c159872e89dc147397771bfc2adf58 (MD5) Seo_Jeehyeon.bib: 26024 bytes, checksum: 96b66eb030dd85236964a00e228ac3f0 (MD5) license.txt: 4058 bytes, checksum: de87d36988eb011e72664524590606ed (MD5) Seo_Jeehyeon.tex: 114824 bytes, checksum: 33639eed0e7fc153244249217d0bd826 (MD5) http://hdl.handle.net/2142/24329 en Copyright 2011 Jeehyeon Seo Bi-Lipschitz uniformly perfect Coloring map Whitney decomposition the Grushin plane singular sub-Riemannian manifold A characterization of Bi-Lipschitz embeddable metric spaces in terms of local Bi-Lipschitz embeddability Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD