Short··0:12·Convex and metric geometry
ChatGPT's Creators Say They Proved This | Mahler's Conjecture
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About this video
The symmetric Mahler conjecture: the volume product of an origin-symmetric convex body and its polar is at least 4^n/n!. OpenAI reports an all-dimensional result in its September 22, 2026 manuscript, released in October. This animation illustrates a three-dimensional family of conjugate Lp balls; the three-dimensional theorem was already known. It is not a visualization of the proof or an independent verification.
The gold and blue solids use the same spatial units. A cube and its octahedral polar exchange roles through a sphere pair. Their volume product ranges from 32/3 to 16*pi^2/9.
Manuscript: github.com/openai/math/blob/main/preprints/The-symmetric-Mahl…
Release: openai.com/index/sharing-ai-progress-in-mathematics/
Original Blender animation and original music. #Mathematics #Geometry #OpenAI
About the problem
Convex and metric geometry·OpenAI Math·Result 087
The Mahler conjectures, functional inequalities and polar-product symplectic width
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product K× K° in dimension 2n has Gromov width 4. (Lean)
Topics
Mahler conjecturesymmetric Mahler conjectureMahler's conjectureMahler conjecture explainedpolar dualityconvex geometryvolume productmathematicsmath visualization3D mathOpenAI mathematicsChatGPTBlender animation
