Almost bi-Lipschitz embeddings and almost homogeneous sets

Olson, Eric; Robinson, James

doi:10.1090/S0002-9947-09-04604-2

Almost bi-Lipschitz embeddings and almost homogeneous sets
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by Eric J. Olson and James C. Robinson

Trans. Amer. Math. Soc. 362 (2010), 145-168

DOI: https://doi.org/10.1090/S0002-9947-09-04604-2

Published electronically: 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 $X$ 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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