Short··0:22·Real and complex analysis
Kakeya Conjecture in 3D and 4D: OpenAI Claims a Proof
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About this video
A Kakeya set contains a unit needle pointing in every direction. Besicovitch showed such a set can have zero volume, and the Kakeya conjecture says it is still full-dimensional; in 3D the dimension statement was proved in 2025. OpenAI claims proofs that every four-dimensional Kakeya set has Hausdorff dimension four, and of the stronger Kakeya maximal conjecture in three dimensions; these are OpenAI's preprints, not yet peer reviewed.
Papers (OpenAI, catalogue result 074):
- The Kakeya maximal conjecture in three dimensions: github.com/openai/math/blob/main/preprints/The-Kakeya-maximal…
- Every four-dimensional Kakeya set has full Hausdorff dimension: github.com/openai/math/blob/main/preprints/Every-four-dimensi…
The animation illustrates the result; it is not a proof.
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Every result explained: gradientdamage.com/p/074
About the problem
Real and complex analysis·OpenAI Math·Result 074
Kakeya in three and four dimensions
Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps L3(ℝ3) to L3(S2) with norm Oε(δ-ε) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
Topics
Kakeya conjectureKakeya setKakeya needle problemBesicovitch setKakeya maximal conjectureHausdorff dimensionharmonic analysisgeometric measure theoryPerron treeWang Zahl4D geometryfourth dimensiontesseractOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsgeometrymath shorts3D animationGradient Damage
