Proof of the Kakeya conjecture in three dimensions
This page delivers no verdict; it presents claims and evidence with their sources. No theory in science is absolute. Method
For everyone
Sets that contain a unit line segment in every direction (Kakeya sets) can have zero volume. Even so, the conjecture says they are as large, in "dimension", as the space they live in. Hong Wang and Joshua Zahl posted a proof of the three-dimensional case to arXiv in February 2025.
For specialists
Wang and Zahl show that for families of δ-tubes in ℝ³ in which not too many tubes lie inside a common convex set, the union has almost maximal volume. Consequently every Kakeya set in ℝ³ has Minkowski and Hausdorff dimension 3. The two-dimensional case was settled by Davies (1971). For n ≥ 4 the conjecture remains open. The result connects to the restriction and Bochner–Riesz conjectures in harmonic analysis.
Claims
- Every Kakeya set in ℝ³ has Hausdorff and Minkowski dimension 3.[1]
Evidence
Open questions
- The Kakeya conjecture in four and higher dimensions is still open.
Connections
Sources
- Preprint (2025). Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensionsarXiv:2502.17655archive
- Web (2025). The three-dimensional Kakeya conjecture, after Wang and Zahl. What's new (blog)terrytao.wordpress.comarchive
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Cite this page
Athena contributors. (2026). Proof of the Kakeya conjecture in three dimensions. Athena: The tree of human knowledge. Retrieved November 1, 2026, from https://latifkedi.github.io/athena/en/d/kakeya/
@misc{athena_kakeya,
title = {Proof of the Kakeya conjecture in three dimensions},
author = {{Athena contributors}},
year = {2026},
howpublished = {\url{https://latifkedi.github.io/athena/en/d/kakeya/}},
note = {Athena: The tree of human knowledge. Accessed: 2026-11-01}
}Added: September 27, 2026 · Last changed: September 27, 2026 · Page history