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Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space

OpenAI · September 27, 2026

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Abstract

We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least $\sqrt{1+t^2/12}-1$, and the countably branching diamond of depth k has distortion at least $\sqrt{1+k/12}$ in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.

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Formalization

Possibly relevant literature (found by the agent, not checked)

  • F. Baudier, N. J. Kalton, G. Lancien, A new metric invariant for Banach spaces (arXiv:0912.5113; Studia Math. 199 (2010) 73–94). Characterizes metric embeddability of the countably branching hyperbolic tree via Szlenk indices of $X$ and $X^*$, with a converse for reflexive spaces; this is the tree counterpart of the diamond question that Theorem 1.1 answers negatively for a reflexive non-AUC-renormable space.

Elsewhere

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Cite

@misc{OAI:Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026,
  author = {{OpenAI}},
  title = {{Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf}{OAI:Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-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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