Metrical characterization of super-reflexivity and linear type of Banach spaces

Abstract

Abstract. We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of superreflexivity in Banach spaces in terms of uniform embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with the metrics dp induced by the p norms. Mathematics Subject Classification (2000). 46B20, 51F99.

Best constants for Lipschitz embeddings of metric spaces into c0

“…Let us also mention that we do not know whether the metric spaces c 0 and c + 0 are Lipschitz isomorphic. Extending a work of J. Bourgain [7], F. Baudier recently proved in [4] that the infinite dyadic tree equipped with the geodesic distance metrically embeds into a Banach space X if and only if X is not super-reflexive. Together with the second named author, F. Baudier also showed in [5] that any locally finite metric space metrically embeds into any Banach space without cotype.…”

Tight Embeddability of Proper and Stable Metric Spaces

“…Does an analogue result hold where L∞ and n ∞ 's are replaced by Lp and n p 's, respectively? The construction in [5] combines a gluing technique introduced in [3], that is tailored for locally nite metric spaces (as shown in [6], and ultimately in [26]), together with a net argument suited for proper spaces, whose inspiration goes back to [7] Proposition 7.18. Due to the technicality of this construction, it is di cult and unclear how to make any progress towards the two problems.…”

On embeddings of locally finite metric spaces into ℓ

“…Let A be a locally finite metric space whose finite subsets admit bilipschitz embeddings with uniformly bounded distortions into a Banach space X. Then, A also admits a bilipschitz embedding into X. Theorem 1.3 has many predecessors, see [2,3,4,20,21]. Applications of this theorem to the coarse embeddings important for Geometric Group Theory and Topology are discussed in [22].…”