The Kakeya conjecture in three dimensions
A Kakeya set (or Besicovitch set) in is a set that contains a unit line segment in every direction. Such sets can have volume zero. Prove that every Kakeya set in nevertheless has Hausdorff dimension 3.
History and references
Partly solved.
In 1917 Sōichi Kakeya asked how small a region can be in which a needle can be turned all the way around. Abram Besicovitch showed in the 1920s that a set containing a unit segment in every direction can have measure zero. The modern conjecture says such sets are still as large as possible in the sense of dimension, and it is closely tied to central problems in harmonic analysis such as the restriction conjecture. In February 2025 Hong Wang and Joshua Zahl posted a 127-page proof of the three-dimensional case. Experts have studied and streamlined the argument without finding errors, and Wang received a Fields Medal in 2026 partly for this work, but as of this writing the paper is still listed on arXiv without a journal reference, so it is marked here as a proof announced and under review. The conjecture in four and more dimensions remains open.
References
Solutions 0
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