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

Q: What type of study is this?

This is a Quantitative Study study.

Hong Wang; Joshua Zahl

doi:10.48550/arxiv.2502.17655

February 24, 2025Open Access

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

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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.

Implication

This analysis shows the union of tubes has near-maximal volume in R^3, suggesting properties of Kakeya sets.

Abstract

We study sets of δ 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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Cite This Study

Wang et al. (2025) studied this question.

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

View Full Paper

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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