A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

The main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low-distortion embeddings into all non-superreflexive spaces. This method is based on the theory of equal-signs-additive sequences developed by Brunel and Sucheston (1975-1976). We also show that some of the low-distortion embeddability results obtained using this method cannot be obtained using the method based on the factorization between the summing basis and the unit vector basis of ${\mathrm{\ell }}_1$, which was used by Bourgain (1986) and Johnson and Schechtman (2009).