[Paper Review] Embeddings of proper metric spaces into Banach spaces

Baudier Florent|arXiv (Cornell University)|Jun 19, 2009

Advanced Banach Space Theory10 references3 citations

TL;DR

This paper establishes that every proper metric space admits a strong uniform embedding into any Banach space without cotype, and further proves that proper subsets of Lp-spaces coarsely bi-Lipschitz embed into Banach spaces uniformly containing ℓp^n. A converse result shows that if all proper metric spaces embed with controlled distortion into a Banach space, then the space must lack nontrivial cotype, characterizing such spaces via embedding properties.

ABSTRACT

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use this locally finite result to construct a coarse bi-Lipschitz embedding for proper subsets of any $\mathcal{L}_p$-space into any Banach space $X$ containing the $\ell_p^n$'s. Finally using an argument of G. Schechtman we prove that for general proper metric spaces and for Banach spaces without cotype a converse statement holds.

Motivation & Objective

• To characterize Banach spaces that admit strong uniform embeddings of all proper metric spaces.
• To investigate coarse bi-Lipschitz embeddability of proper subsets of Lp-spaces into Banach spaces containing ℓp^n uniformly.
• To establish a converse statement: if all proper metric spaces embed with controlled distortion, then the target Banach space must lack nontrivial cotype.
• To extend known results on Lipschitz embeddings of locally finite metric spaces into cotype-free Banach spaces to the coarse and uniform setting.
• To use ultra-product techniques and duality arguments to derive linear embeddability in ultrapowers, linking nonlinear embeddings to linear structure.

Proposed method

• Constructing strong uniform embeddings via maximal nets and metric entropy estimates in proper metric spaces.
• Using the fact that Banach spaces without cotype uniformly contain ℓ∞^n to build embeddings via finite-dimensional approximation.
• Applying a local embedding result for locally finite subsets of Lp-spaces into Banach spaces containing ℓp^n uniformly.
• Composing a coarse bi-Lipschitz map from the metric space to a finite net with a Lipschitz embedding of the net into the target space.
• Using ultra-product techniques to lift coarse embeddings into linear embeddings in ultrapowers, leveraging Heinrich-Mankiewicz differentiability arguments.
• Proving the converse via a contradiction argument: assuming such embeddings exist for all proper spaces forces the target space to have no nontrivial cotype.

Experimental results

Research questions

• RQ1 Can every proper metric space be strongly uniformly embedded into any Banach space without cotype?
• RQ2 Under what conditions does a Banach space admit coarse bi-Lipschitz embeddings of all proper subsets of Lp-spaces?
• RQ3 Is the absence of nontrivial cotype necessary for a Banach space to coarsely embed all proper metric spaces with controlled distortion?
• RQ4 Can the embedding properties of a Banach space be characterized by its ability to embed all proper metric spaces with uniform distortion?
• RQ5 To what extent do coarse or uniform embeddings into a Banach space imply linear structure in its ultrapower or dual?

Key findings

• Every proper metric space admits a strong uniform embedding into any Banach space without cotype.
• Proper subsets of Lp-spaces coarsely bi-Lipschitz embed into any Banach space uniformly containing ℓp^n, with universal distortion and additive constants depending only on the Lp-structure.
• A Banach space X has no nontrivial cotype if and only if every proper metric space admits a coarse bi-Lipschitz embedding into X with uniform distortion and additive constants.
• The converse of the local finite embedding result holds: if all locally finite metric spaces embed into X with uniform distortion, then X must uniformly contain ℓ∞^n, i.e., have no nontrivial cotype.
• Using ultra-product techniques and duality, the paper shows that ℓ∞^n embeds linearly into the second dual of the ultrapower of X, linking nonlinear embedding properties to linear structure.
• The argument of G. Schechtman establishes that the existence of a universal coarse bi-Lipschitz embedding constant for all proper metric spaces characterizes Banach spaces without cotype.