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

TL;DR

This paper introduces a novel embedding technique for finite metric spaces into non-superreflexive Banach spaces, leveraging equal-signs-additive sequences, and demonstrates its advantages over previous methods.

Contribution

The paper develops a new embedding method based on equal-signs-additive sequences, enabling low-distortion embeddings into non-superreflexive spaces, surpassing prior approaches.

Findings

New embedding method based on equal-signs-additive sequences

Some embeddings cannot be achieved by previous factorization methods

Demonstrates broader applicability of low-distortion embeddings

Abstract

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 $\ell_1$, which was used by Bourgain (1986) and Johnson and Schechtman (2009).