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 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 are almost bi-Lipschitz between and its image.