University of Illinois Urbana-Champaign

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

Seo, Jeehyeon

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

Description

Title
A characterization of Bi-Lipschitz embeddable metric spaces in terms of local Bi-Lipschitz embeddability
Author(s)
Seo, Jeehyeon
Issue Date
2011-05-25T15:02:38Z
Director of Research (if dissertation) or Advisor (if thesis)
Tyson, Jeremy T.
Doctoral Committee Chair(s)
Wu, Jang-Mei
Committee Member(s)
  • Tyson, Jeremy T.
  • D'Angelo, John P.
  • Merenkov, Sergiy A.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2011-05-25T15:02:38Z
Keyword(s)
  • Bi-Lipschitz
  • uniformly perfect
  • Coloring map
  • Whitney decomposition
  • the Grushin plane
  • singular sub-Riemannian manifold
Abstract
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.
Graduation Semester
2011-05
Permalink
http://hdl.handle.net/2142/24329
Copyright and License Information
Copyright 2011 Jeehyeon Seo

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