332. Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension

Metric Markov cotype of ℓ₁ and Hilbert-space Lipschitz extension

1 paper

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.

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First page of Metric Markov Cotype Two of <i>ℓ</i><sub>1</sub>

Metric Markov Cotype Two of ℓ₁

OpenAI · October 5, 2026

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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@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}
}

Metric Markov cotype of ℓ₁ and Hilbert-space Lipschitz extension

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Papers 1

Metric Markov Cotype Two of ℓ₁

Formal verification

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Papers 1

Metric Markov Cotype Two of ℓ1

Formal verification

Metric Markov Cotype Two of ℓ₁