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Midpoint convexity from bounded tree potentials and path costs

OpenAI · September 27, 2026

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Abstract

Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound $\widehat\delta(t)\ge\sqrt{1+t^2/4}-1$ for $0\lt t\lt 1$, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.

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Cite

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

Lean formalization

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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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