Small distortion and volume preserving embeddings for planar and Euclidean metrics

A finite metric space, (S,d) , contains a finite set of points and a distance function on pairs of points. A contraction is an embedding, h, of a finite metric space (S, d) into Rd where for any u, v E S, the Euclidean (&) distance between h(u) and h(v) is no more than d(u, v). The distortion of the embedding is the maximum over pairs of the ratio of d(u, w) and the Euclidean distance between h(u) and h(v).