Almost bi-Lipschitz embeddings and almost homogeneous sets

Details Almost bi-Lipschitz embeddings and almost homogeneous sets Almost bi-Lipschitz embeddings and almost homogeneous sets

2007-05-03

arxiv.org/abs/0705.0424v1

Trans AMS 362 (2010) 145-168

Mathematics

Metric Geometry

Submitted to Transactions of the AMS

Scientific paper

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. In fact we show that if $X$ is a compact subset of a Banach space and $X-X$ is almost homogeneous then, for $N$ sufficiently large, a prevalent set of linear maps from $X$ into $\Re^N$ are almost bi-Lipschitz between $X$ and its image. We are then able to use the Kuratowski embedding of $(X,d)$ into $L^\infty(X)$ to prove a similar result for compact metric spaces.

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