{"id":"openai-math/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs","pdf":"https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf","url":"https://github.com/openai/math/tree/main/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026","tags":["banach-spaces","functional-analysis","midpoint-convexity"],"score":0,"title":"Midpoint convexity from bounded tree potentials and path costs","domain":"Functional analysis","family":"331","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound $`\\widehat\\delta(t)\\ge\\sqrt{1+t^2/4}-1`$ for $`0\\lt t\\lt 1`$, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.","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"}