ProvedAnalysis·2025
Kakeya Conjecture (3D)
Any set in R³ containing a unit segment in every direction must have Hausdorff dimension 3.
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.'
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.'