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Metric Markov Cotype Two of ℓ1
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Abstract
We prove that the real Banach space ℓ1 has metric Markov cotype two, answering a question of Mendel and Naor. As a consequence, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole Hilbert space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
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Notes
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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.
Related prior literature mentioned by others
- Manor Mendel and Assaf Naor, Spectral calculus and Lipschitz extension for barycentric metric spaces (arXiv:1301.3963, 2013). The Latent Seminar exposition identifies the Mendel–Naor extension theorem as the additional input used to deduce the manuscript's Hilbert-to-ℓ₁ Lipschitz extension consequence from metric Markov cotype two.
Possibly relevant literature (found by the agent, not checked)
- Assaf Naor, De-Höldering factorization (arXiv:2609.07564, v1 7 Sep 2026, v2 10 Sep 2026). Its abstract states that every L₁-valued Lipschitz function on a subset of ℓ₂ extends to all of ℓ₂, answering Ball's 1992 question, and its added-in-proof note (HTML v2) announces forthcoming work with M. Mendel showing that L₁ has metric Markov cotype 2. Both bear on the paper's main theorem and its Lipschitz extension corollary; the preprint predates the manuscript and is not cited in it.
- Sebastian Gietl, Lipschitz extensions from spaces of nonnegative curvature into CAT(1) spaces (arXiv:2408.00564, 2024; v3 2025). Proves metric Markov cotype 2 for small-radius CAT(κ) spaces in the Mendel–Naor framework that the paper's definition and extension corollary use, giving background on which spaces were already known to have metric Markov cotype 2.
Digests and expositions
- Latent Seminar, Metric Markov cotype two of ℓ₁ (6 October 2026). Explains family 332's cut representation, cubic smoothing, quartic martingale potential and expected-median construction; the source states it covers statements and selected proof mechanisms rather than an independent verification of the complete proof. [family-level]
- Open Mathematics Courses, Metric Markov cotype of ℓ₁ (undated). Three lessons with proofs and exercises develop the manuscript's geometric-time comparison, weighted cuts, flat cubic and stopped-walk construction; the source credits writing and self-checking to Claude Opus 5.5 and says independent review is incomplete. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 332
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Metric-Markov-Cotype-Two-of-l1-October-5-2026,
author = {{OpenAI}},
title = {{Metric Markov Cotype Two of $\ell_1$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf}{OAI:Metric-Markov-Cotype-Two-of-l1-October-5-2026}},
year = {2026}
}Family 332: Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension
Proves that real ℓ1 has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
This is the only paper in this family.