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ProvedAnalysis·2025

Kakeya Conjecture (3D)

Any set in R³ containing a unit segment in every direction must have Hausdorff dimension 3.

Formula

Kakeya theorem (3D)
K⊂R3,  K Besicovitch set⟹dim⁡HK=dim⁡‾MK=3K\subset\mathbb{R}^3,\; K\text{ Besicovitch set}\quad\Longrightarrow\quad \dim_{\mathrm H}K=\overline{\dim}_{\mathrm M}K=3

Wang and Zahl proved in 2025 that every Kakeya set in three dimensions has full Hausdorff and Minkowski dimension.

Neighborhood-volume form
Kδ={x:dist⁡(x,K)<δ},dim⁡‾MK=3−lim inf⁡δ↓0log⁡∣Kδ∣log⁡δK_\delta=\{x:\operatorname{dist}(x,K)<\delta\},\quad \overline{\dim}_{\mathrm M}K=3-\liminf_{\delta\downarrow0}\frac{\log |K_\delta|}{\log\delta}

The full-dimensional Minkowski statement can be read through the decay rate of δ-neighborhood volume.

Wolff's hairbrush bound
dim⁡HK≥52for K⊂R3\dim_H K \ge \tfrac{5}{2}\quad\text{for }K\subset\mathbb{R}^3

Wolff (1995) established dim ≥ 5/2 using the hairbrush argument, the first major advance beyond Bourgain's earlier bound.

Summary

The Kakeya conjecture asks: what is the smallest possible Hausdorff dimension of a compact set in Rⁿ that contains a unit line segment in every direction? In 3D, the conjecture predicts dimension 3 (full). This was open for 50 years until Wang and Zahl's proof in February 2025, which Quanta Magazine called 'once in a century.'

Sources

Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

arXiv:2502.176552025

Kakeya sets in R^3

1999

A proof of the Kakeya set conjecture over rings of integers modulo square-free N

arXiv:2406.198032025

Videos

Thumbnail for A Once-in-a-Century Proof: The Kakeya Conjecture

A Once-in-a-Century Proof: The Kakeya Conjecture

Quanta Magazine
Thumbnail for Kakeya's Needle Problem - Numberphile

Kakeya's Needle Problem - Numberphile

Numberphile
Thumbnail for The Kakeya needle problem (the squeegee approach)

The Kakeya needle problem (the squeegee approach)

Mathologer