# Kakeya conjecture in three dimensions

- kind: problem
- status: solved (evidence tier V3)
- year: 1917

**Statement.** Every set in $\mathbb{R}^3$ containing a unit segment in every direction has Hausdorff dimension 3.

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](/wiki/Problem:kakeya-general.md); the argument for the link has not been written yet.

## Sources

- [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set)

[Geometry and topology](/wiki/Space:geometry-topology.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- special case of [kakeya-general](/wiki/Problem:kakeya-general.md) — 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.

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