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arXiv · 2502.17655

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

Hong Wang·Joshua Zahl
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Abstract

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.

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Hong Wang, Joshua Zahl. 2025-02-24. Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions. https://arxiv.org/abs/2502.17655

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