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Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0
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Abstract
We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to $Z\oplus_\infty c_0$. The same space contains a bi-Lipschitz image of every separable metric space.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 324 (lean/docs/324.md, 2026-10-06). The release's Lean scope document says the formalized result builds a separable real Banach space bi-Lipschitz equivalent to its product with $c_0$, with no closed subspace isomorphic to $c_0$ and bi-Lipschitz universal for separable metric spaces; it points to the comparator file
C0Absorption.lean. [family-level]
Possibly relevant literature (found by the agent, not checked)
- N. J. Kalton and G. Lancien, Best constants for Lipschitz embeddings of metric spaces into c_0 (arXiv:0708.3924, 2007). Proves that every separable metric space Lipschitz 2-embeds into $c_0$ and that 2 is sharp, improving Aharoni's theorem, which the paper uses to show that $Z$ is bi-Lipschitz universal.
- F. Baudier, G. Lancien, P. Motakis and Th. Schlumprecht, Coarse and Lipschitz universality (arXiv:2004.04806, 2020). Introduces a Lipschitz $c_0$-index that detects bi-Lipschitz copies of $c_0$ in Polish spaces and gives criteria for Lipschitz universality, which relates to the bi-Lipschitz copy of $c_0$ and the universality corollary for $Z$.
- F. P. Baudier and G. Lancien, Asymptotic and nonlinear geometries of Banach spaces and their interactions (arXiv:2512.00817, 2025-11-30). Book-length survey of how linear asymptotic structure behaves under bi-Lipschitz, coarse-Lipschitz, coarse and uniform embeddings, the setting of Kalton's Problem 2 that Theorem 1.1 addresses.
Digests and expositions
- Latent Seminar, Bi-Lipschitz equivalence without linear isomorphism (2026-10-06). Exposition of both constructions' triangular inverse identities and distinct linear obstructions; it states explicitly that it does not independently verify the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 324
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026,
author = {{OpenAI}},
title = {{Bi-Lipschitz Absorption of $c_0$ Without a Linear Copy of $c_0$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf}{OAI:Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/324.md. The Lean statements have not been checked by people against the paper's claims.
Family 324: Lipschitz equivalent Banach spaces need not be linearly isomorphic
Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.
Other papers in this family:
- Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic · September 24, 2026