A characterization of Bi-Lipschitz embeddable metric spaces in terms of local Bi-Lipschitz embeddability

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Seo, Jeehyeon

Contributors dc:contributor

• Tyson, Jeremy T.
• Wu, Jang-Mei
• D'Angelo, John P.
• Merenkov, Sergiy A.

• Bi-Lipschitz
• uniformly perfect
• Coloring map
• Whitney decomposition
• the Grushin plane
• singular sub-Riemannian manifold

Statement dc:rights

• Copyright 2011 Jeehyeon Seo

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/24329

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/24329