Kakeya set conjecture in higher dimensions (n ≥ 4)

Establish that every Kakeya set in Euclidean space R^n (for dimensions n ≥ 4) has Minkowski and Hausdorff dimension n, i.e., prove the full Kakeya set conjecture beyond the planar and three-dimensional cases.

Background

The paper proves the Kakeya set conjecture in R3, showing every Kakeya set in three dimensions has full dimension. Historically, Davies proved the conjecture in the plane (n=2), leaving higher dimensions unresolved. The authors note the general conjecture was open in three and higher dimensions; their work settles n=3, so the remaining open cases are n ≥ 4.

Resolving the conjecture in higher dimensions would complete one of the central problems in geometric measure theory and harmonic analysis, with wide-ranging consequences for restriction theory and related maximal inequalities.

References

This conjecture was proved by Davies when $n=2$, and is open in three and higher dimensions.

— Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions (2502.17655 - Wang et al., 24 Feb 2025) in Introduction, Section 1