oai-math-revised beta

Browse › Functional analysis › 331 Reflexive midpoint convexity and diamond distortion

Midpoint convexity from two recursive potentials

OpenAI · September 27, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least $t^3/128$ for $0\lt t\lt 1$, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

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

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Midpoint-convexity-from-two-recursive-potentials-September-27-2026,
  author = {{OpenAI}},
  title = {{Midpoint convexity from two recursive potentials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf}{OAI:Midpoint-convexity-from-two-recursive-potentials-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.

Other papers in this family: