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

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 nonsuperreflexive spaces. This method is based on the theory of equal-signs-additive sequences developed by 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 1 , which was used by Bourgain (1986) and Johnson and Schechtman (2009).

A Coding of Bundle Graphs and Their Embeddings Into Banach Spaces

“…Once W ′′ is found, we can use Theorem 5.4 to find a bound on the worst distortion for a bi-Lipschitz embedding of T W ′′ ,κ into a Banach space with an ESA basis. In particular, we'll show that the distortion bound found in Theorem 5.4 is no worse for T W,κ ⊘ T W,κ than it was for T W,κ , allowing us to generalize the characterizations of superreflexifity found in [7] and [9].…”

“…In [4], J. Bourgain proved that the notion of superreflexivity in Banach spaces can be characterized by the non-equi-bi-Lipschitz embeddability of the family of binary trees with finite height. Since then, the non-equi-bi-Lipschitz embeddability of several other families of graphs have also been shown to characterize superreflexivity ( [2], [7], [9]). In [3], F. Baudier et al proved that the non-equi-bi-Lipschitz embeddability of the family of ℵ 0 -branching diamond graphs characterizes the asymptotic uniform convexifiability of refexive Banach spaces with an unconditional asymptotic structure.…”

“…The families of graphs used in [7], [9], and [3] are all contained in a larger class of graphs, called the "bundle graphs". The goal of this paper is to generalize the results mentioned above to this larger class while providing simpler proofs.…”

Lipschitz-free Spaces on Finite Metric Spaces

“…Hence |V (D n )| = 2(1 + n−1 i=0 4 i ). The next special case of the general Definition 1.6, whose metric geometry was studied in [42,50], corresponds to the case where B = K 2,n , and the vertices in the part containing 2 vertices play the roles of the top and the bottom. The usual definition is the following.…”

A characterization of superreflexivity through embeddings of lamplighter groups

“…Several different sets of test-spaces for superreflexivity are known: (1) binary trees [4,14,2,9], (2) binary diamond and Laakso graphs [8,15], (3) multibranching diamond and Laakso graphs [24]. See [23] for a survey on this matter written in 2014.…”