Tight Embeddability of Proper and Stable Metric Spaces

Abstract

Abstract:We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [ , ∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the n p 's. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of re exive Banach spaces.

Almost bi-Lipschitz embeddings and proper subspaces of a Banach space -- An extension of a theorem by M.I. Ostrovskii

“…In fact, we sharpen a result obtained by M.I. Ostrovskii in the case of a bi-Lipschitz embeddability for an almost bi-Lipschitz embeddability (notion introduced in [2]).…”

“…Here, the goal is, under a condition of crude finite representability of a Banach space X in each finite codimensional subspace of another Banach space Y , to obtain an almost bi-Lipschitz embeddability of the proper subspaces of X into Y . Two existing theorems inspired the central theorem of this article (one from [7] and the other from [2]).…”

“…Otherwise, in 2015, F. Baudier et G. Lancien proved in [2], among others, that, for p ∈ [1, +∞] and Y a Banach space which contains uniformly the ℓ n p spaces (you will find the definition later), with M a proper subset of L p , M almost bi-Lipschitzly embeds into Y .…”

The coarse geometry of Tsirelson’s space and applications

“…There is rather strong evidence that ℓ 2 is the space into which it is the most difficult to embed. It was shown in [33] that every locally finite metric subset of ℓ 2 admits a bi-Lipschitz embedding into every infinite dimensional Banach space, and in [6] that every proper subset of ℓ 2 (i.e. whose closed balls are compact) is almost Lipschitz embeddable into every infinite dimensional Banach space.…”

Book Review: Metric embeddings: bilipschitz and coarse embedddings into Banach spaces

“…In order to improve the easier direction in Bourgain's characterization of superreflexivity, the barycentric gluing technique was introduced in [8]. This technique, inspired by the work of Ribe [60], was fruitfully implemented in various situations (see for instance [9]) and had some unexpected consequences for the geometry of infinite metric spaces. In particular it is a crucial ingredient in the proof of Ostrovskii's finite determinacy of the bi-Lipschitz embeddability for locally finite metric spaces [56].…”