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Almost bi-Lipschitz embeddings and almost homogeneous sets

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Olson, Eric J. and Robinson, James C. (James Cooper), 1969-. (2010) Almost bi-Lipschitz embeddings and almost homogeneous sets. Transactions of the American Mathematical Society, Vol.362 (No.1). pp. 145-168. ISSN 0002-9947

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Official URL: http://dx.doi.org/10.1090/S0002-9947-09-04604-2

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 R-N are almost bi-Lipschitz between X and its image.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Embeddings (Mathematics), Hilbert space, Lipschitz spaces
Journal or Publication Title:	Transactions of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0002-9947
Date:	January 2010
Volume:	Vol.362
Number:	No.1
Number of Pages:	24
Page Range:	pp. 145-168
Identification Number:	10.1090/S0002-9947-09-04604-2
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Royal Society (Great Britain)
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URI:	http://wrap.warwick.ac.uk/id/eprint/16573