Kakeya set conjecture in Euclidean space

Establish that every Besicovitch set K ⊂ R^n has Minkowski and Hausdorff dimension equal to n, thereby resolving the Kakeya set conjecture in all dimensions.

Background

Besicovitch sets are compact subsets of Euclidean space that contain a unit line segment in every direction. In R^2, Besicovitch constructed a null-measure example, motivating quantitative questions about compressibility of such sets at small scales.

The Kakeya set conjecture posits full dimension for all Besicovitch sets and is a cornerstone problem linked to Fourier analysis and geometric measure theory. It is proved in R^2 (Davies; Córdoba) and, via Wang–Zahl, in R^3, but remains open in higher dimensions.