Transactions of the American Mathematical Society

Author(s): Eric J. Olson; James C. Robinson
Journal: Trans. Amer. Math. Soc. 362 (2010), 145-168.
MSC (2000): Primary 54F45, 57N35
Posted: August 17, 2009
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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

of a Hilbert space

into a finite-dimensional Euclidean space. We show that if

is a compact subset of a Hilbert space and

is almost homogeneous, then, for

sufficiently large, a prevalent set of linear maps from

into $\mathbb{R}^N$ are almost bi-Lipschitz between

and its image.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54F45, 57N35

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: 10.1090/S0002-9947-09-04604-2
PII: S 0002-9947(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
Posted: 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.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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