Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 331-01. Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space
Manuscript map · lines 8775–8784
OpenAI@2026-10-07#rem:oai-p-331-01
Authors: OpenAI. Family 331.
Title: Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space.
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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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