Level 1: Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space

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LEVEL 1 OF 7 · Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming

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

expertly designed by an internal OpenAI model · released 2026-09-27 · original PDF

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How to play: 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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