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kakeya-3d
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A Kakeya set can have zero volume yet still hold a needle in every direction. The conjecture says such sets cannot be small in dimension. The two-dimensional case was done in 1971. Wang and Zahl's 2025 proof of the three-dimensional case is widely considered one of the decade's biggest results in analysis. The case of dimension four and up remains open.
Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.
Closed 2025, Hong Wang and Joshua Zahl.
Related problems
- The original curation records this as a special case of Kakeya set conjecture, all dimensions; the argument for the link has not been written yet.