Distortion in the Finite Determination Result for Embeddings of Locally Finite Metric Spaces Into Banach Spaces

Abstract

Given a Banach space X and a real number alpha >= 1, we write: (1) D(X) <= alpha if, for any locally finite metric space A, all finite subsets of which admit bilipschitz embeddings into X with distortions <= C, the space A itself admits a bilipschitz embedding into X with distortion <= alpha . C; (2) D(X) = alpha(+) if, for every epsilon > 0, the condition D(X) <= alpha + epsilon holds, while D(X) <= alpha does not; (3) D(X) <= alpha(+) if D(X) = alpha(+) or D(X) <= alpha. It is known that D(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D((circle plus(infinity)(n= 1) X-n)(p)) <= 1(+) for every nested family of finite-dimensional Banach spaces {X-n}(n=1)(infinity) and every 1 <= p <= 8 infinity. (2) D((circle plus 8(n=1)(infinity)l(infinity)(n) )(p)) = 1(+) for 1 < p < infinity. (3) D(X) <= 4(+) for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier-Lancien result (2008).