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Midpoint lenses in segment spaces

OpenAI · September 27, 2026

Lean

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Abstract

For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by $2\sqrt{R^2-\|x\|^2}$, where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.

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Cite

@misc{OAI:Midpoint-lenses-in-segment-spaces-September-27-2026,
  author = {{OpenAI}},
  title = {{Midpoint lenses in segment spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf}{OAI:Midpoint-lenses-in-segment-spaces-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/331.md. The Lean statements have not been checked by people against the paper's claims.

Family 331: Reflexive midpoint convexity and diamond distortion

Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least $\sqrt{1+k/12}$, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.

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