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 R^3, 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.