Math Conjectures

Kakeya set conjecture in dimensions n≥4n\ge4

OPENLandmarkConjectureProposed c. 1971 · Canonical special case

Canonical statement

Let n≥4n\ge4 and let E⊂RnE\subset\mathbb R^n be a Borel set such that for every v∈Sn−1v\in S^{n-1} there is xv∈Rnx_v\in\mathbb R^n with {xv+tv:0≤t≤1}⊂E\{x_v+tv:0\le t\le1\}\subset E. Then dim⁡HE=n\dim_{\mathrm H}E=n.
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Let \(n\ge4\) and let \(E\subset\mathbb R^n\) be a Borel set such that for every \(v\in S^{n-1}\) there is \(x_v\in\mathbb R^n\) with \(\{x_v+tv:0\le t\le1\}\subset E\). Then \(\dim_{\mathrm H}E=n\).
ANAL-FA-006:[Wolff1999KakeyaSurvey][WangZahl2025Kakeya3D][Zahl2025KakeyaSurvey][Tao2026KakeyaBourbaki]
Analysis and functional analysisDiscrete geometry

Notes

A Kakeya (Besicovitch) set in Rn\mathbb R^n is a set containing a unit line segment in every direction. The Kakeya set conjecture asserts that such a set, even if it has Lebesgue measure zero, must have full Hausdorff dimension nn. The modern Hausdorff-dimension formulation emerged around the early 1970s; this entry records the cases n≥4n\ge4.

The planar case is classical: Kakeya sets in R2\mathbb R^2 have dimension 22. In higher dimensions a long sequence of partial lower bounds was developed through geometric, combinatorial and polynomial methods [Wolff1999KakeyaSurvey], [Zahl2025KakeyaSurvey]. The three-dimensional case was settled by Wang and Zahl, whose proof proceeds through volume estimates for unions of convex sets [WangZahl2025Kakeya3D]; the argument is the subject of a Bourbaki exposition [Tao2026KakeyaBourbaki].

The stronger Kakeya maximal function inequality, which would imply the set conjecture, is likewise unresolved. Despite the accumulated partial dimension bounds, the Kakeya set conjecture remains open in every dimension n≥4n\ge4.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.