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This is a Quantitative Study study.

October 16, 2025Open Access

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

Key Points

The union of δ tubes in R^3 achieves almost maximal volume under certain constraints.
Every Kakeya set in R^3 is shown to have both Minkowski and Hausdorff dimensions equal to 3.
This research utilizes properties of convex sets to establish volume estimates for unions of tubes.
The findings support broader implications for geometric measure theory and dimensional analysis.

Abstract

We study sets of tubes in R³, 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 R³ has Minkowski and Hausdorff dimension 3.

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Wang et al. (Mon,) studied this question.

www.synapsesocial.com/papers/68f0d5eb105731330a2b1fca — DOI: https://doi.org/10.48550/arxiv.2502.17655

Authors

Hong Wang

Joshua Zahl

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

Key Points

Abstract

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Discussion

Cite this study

Authors

Actions

References and Citations

Citation Network

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Discussion