Bi-Lipschitz embeddings into low-dimensional Euclidean spaces

Let (X, d), (Y, p) be metric spaces and / : X —• Y an infective mapping. We put l i / I U = s u p { ^ y y ) ; x , y € X,x*y}, d(x,y) and dist(f) = ||/||i,t>||/"~|Ut> (the distortion of the mapping / ) . The distortion can be considered as a measure of "faithfulness" of the mapping from a metric point of view. The Lipschitz distance of X from subspaces of Y is defined as follows: dist(X, C V) = inf{dist(f);f : X —> Y an injective mapping}. We investigate the maximum possible Lipschitz distance of a n-point metric space X from subspaces of ^-dimensional Euclidean space E (where k is a small fixed number); let us denote this quantity by / (n , k). We obtain the following bounds: / ( n , * ) = 8 ( n ) (A: = 1,2) f(n, k) = 0(n'*(log n)') (k > 3) /(n,Jfe)=Q(n) (k even), f(n,k) = Q(n<*>) (k odd).