ALMOST BI-LIPSCHITZ EMBEDDINGS AND ALMOST HOMOGENEOUS SETS

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.