Level 3: Distortion of countably branching diamonds from midpoint and tree energies

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LEVEL 3 OF 7 · Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming

Distortion of countably branching diamonds from midpoint and tree energies

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

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How to play: We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion-D embedding of the depth-k diamond satisfies $D^2\ge1+k/4$. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.

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