Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

By Hong Wang, Joshua Zahl. 2025

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We study sets of $\delta$ tubes in $\mathbb{R}^3$, with the property that not too many tubes can be contained inside a common convex set $V$. We show that the union of tubes from such a set must have almost maximal volume. As a consequence, we prove that every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3.

View Paper: https://www.semanticscholar.org/paper/7d54aaed58c36748d82d90f1bcef27051a70ab00