{"id":"openai-math/Midpoint-lenses-in-segment-spaces","pdf":"https://github.com/openai/math/blob/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf","url":"https://github.com/openai/math/tree/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026","tags":["banach-spaces","asymptotic-convexity","forest-duals"],"score":0,"title":"Midpoint lenses in segment spaces","domain":"Functional analysis","family":"331","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"formalized":true,"family_title":"Reflexive midpoint convexity and diamond distortion"}