{"id":"openai-math/Midpoint-convexity-from-two-recursive-potentials","pdf":"https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf","url":"https://github.com/openai/math/tree/main/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026","tags":["banach-spaces","tree-norms","asymptotic-convexity"],"score":0,"title":"Midpoint convexity from two recursive potentials","domain":"Functional analysis","family":"331","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least $`t^3/128`$ for $`0\\lt t\\lt 1`$, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.","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"}