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Almost bi-Lipschitz embeddings and almost homogeneous sets


Authors: Eric J. Olson and James C. Robinson
Journal: Trans. Amer. Math. Soc. 362 (2010), 145-168
MSC (2000): Primary 54F45, 57N35
DOI: https://doi.org/10.1090/S0002-9947-09-04604-2
Published electronically: August 17, 2009
MathSciNet review: 2550147
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Abstract: This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but `almost homogeneous'. We therefore study the problem of embedding an almost homogeneous subset $ X$ of a Hilbert space $ H$ into a finite-dimensional Euclidean space. We show that if $ X$ is a compact subset of a Hilbert space and $ X-X$ is almost homogeneous, then, for $ N$ sufficiently large, a prevalent set of linear maps from $ X$ into $ \mathbb{R}^N$ are almost bi-Lipschitz between $ X$ and its image.


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Additional Information

Eric J. Olson
Affiliation: Department of Mathematics/084, University of Nevada, Reno, Nevada 89557
Email: ejolson@unr.edu

James C. Robinson
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: j.c.robinson@warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-09-04604-2
Keywords: Assouad dimension, Bouligand dimension, doubling spaces, embedding theorems, homogeneous spaces
Received by editor(s): September 15, 2005
Received by editor(s) in revised form: May 27, 2007
Published electronically: August 17, 2009
Additional Notes: The second author was a Royal Society University Research Fellow when this paper was written, and he would like to thank the Society for all their support. Both authors would like to thank Professor Juha Heinonen for his helpful advice, and Eleonora Pinto de Moura for a number of corrections after her close reading of the manuscript.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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