Level 2: Midpoint lenses in segment spaces

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LEVEL 2 OF 7 · Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming

Midpoint lenses in segment spaces

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

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How to play: For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by $2\sqrt{R^2-\|x\|^2}$, where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.

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