Almost bi-Lipschitz embeddings and almost homogeneous sets

Abstract

This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be em- bedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic corrections). The image of this set is no longer homo- geneous, but 'almost homogeneous'. We therefore study the problem of em- bedding 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 R N are almost bi-Lipschitz between X and its image. We are then able to use the Kuratowski embedding of (X, d) into L ∞ (X) to prove a similar result for compact metric spaces.