[Paper Review] Coarse embeddability into Banach spaces

Mikhail I. Ostrovskii|ArXiv.org|Feb 25, 2008

Advanced Operator Algebra Research49 references19 citations

TL;DR

This paper investigates coarse embeddability of metric spaces into Banach spaces, particularly focusing on whether spaces with bounded geometry can embed coarsely into Hilbert spaces or other Banach spaces. It establishes that coarse non-embeddability into ℓ₂ implies the presence of expander-like structures and demonstrates that ℓ₂ is not the most difficult space to embed into, providing new constructions and estimates for Lipschitz constants in embeddings of ℓ₁ into certain Banach spaces with Schauder decompositions.

ABSTRACT

The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into $\ell_2$ implies presence of expander-like structures? (b) To what extent $\ell_2$ is the most difficult space to embed into?

Motivation & Objective

• To survey the theory of coarse embeddability of metric spaces into Banach spaces, especially for spaces with bounded geometry.
• To investigate whether coarse non-embeddability into ℓ₂ implies the presence of expander-like structures.
• To determine to what extent ℓ₂ is the most difficult Banach space for coarse embeddings.
• To present new results on Lipschitz embeddings of ℓ₁ into Banach spaces with Schauder decompositions into finite-dimensional subspaces.
• To analyze the role of Dvoretzky’s theorem and basic sequence constructions in embedding infinite-dimensional Banach spaces.

Proposed method

• Uses coarse embeddings defined via control functions ρ₁ and ρ₂ satisfying ρ₁(d(x,y)) ≤ d(f(x),f(y)) ≤ ρ₂(d(x,y)) with limᵣ→∞ρ₁(r) = ∞.
• Applies the concept of bounded geometry and locally finite metric spaces to analyze embeddability obstructions.
• Employs constructions based on P. Enflo’s work and adapts them to show non-embeddability of certain locally finite spaces into Hilbert spaces.
• Uses expanders—families of finite, regular, connected graphs with bounded away from zero Cheeger constants—as obstructions to coarse embeddability into ℓₚ for 1 ≤ p < ∞.
• Constructs a Lipschitz map φ from ℓ₁ into a Banach space Y with a Schauder decomposition into ℓ₂ⁿᵢ subspaces, estimating its Lipschitz constant and that of its inverse.
• Applies Dvoretzky’s theorem and basic sequence techniques to show that any infinite-dimensional Banach space contains subspaces isomorphic to spaces with such decompositions.

Experimental results

Research questions

• RQ1 Does coarse non-embeddability into ℓ₂ imply the presence of expander-like structures in the metric space?
• RQ2 To what extent is ℓ₂ the most difficult Banach space for coarse embeddings of metric spaces with bounded geometry?
• RQ3 Can every infinite-dimensional Banach space contain a subspace isomorphic to a space with a Schauder decomposition into finite-dimensional ℓ₂ⁿᵢ subspaces?
• RQ4 What are the sharp estimates for the Lipschitz constants of embeddings from ℓ₁ into Banach spaces with specific Schauder decompositions?
• RQ5 Can the non-embeddability of certain locally finite metric spaces into Hilbert spaces be characterized via expanders?

Key findings

• Coarse non-embeddability into ℓ₂ implies the presence of expander-like structures, as shown by Gromov’s observation that spaces containing isometric copies of expanders cannot embed coarsely into ℓₚ for 1 ≤ p < ∞.
• The space ℓ₂ is not the most difficult space to embed into; there exist Banach spaces with Schauder decompositions into ℓ₂ⁿᵢ subspaces into which certain metric spaces embed coarsely.
• A Lipschitz embedding φ from ℓ₁ into a Banach space Y with Schauder decomposition {Yᵢ} ≅ ℓ₂ⁿᵢ is constructed, with Lipschitz constant estimates derived from norm comparisons involving ||a||_Z and ||b||_Z.
• The inverse of the embedding φ has controlled Lipschitz constant when ||a - b||_Z is much larger than ||a||_Z - ||b||_Z, particularly when ||a - b||_Z ≥ 5(||a||_Z - ||b||_Z).
• For cases where ||a||_Z - ||b||_Z ≥ ||a - b||_Z / 5, the sum of norm contributions in the embedding provides a lower bound of ||a - b||_Z / 5 on ||φ(a) - φ(b)||_Y.
• Using Dvoretzky’s theorem and basic sequence techniques, it is shown that any infinite-dimensional Banach space contains a subspace isomorphic to a space with a Schauder decomposition into finite-dimensional ℓ₂ⁿᵢ subspaces, enabling coarse embeddings into such spaces.