{"id":"openai-math/A-positive-solution-to-Tingleys-problem","pdf":"https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026","tags":["functional-analysis","banach-spaces","isometries"],"score":0,"title":"A positive solution to Tingley’s problem","domain":"Functional analysis","family":"322","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.FA","summary":"Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"Tingley’s sphere-isometry problem"}