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

TL;DR

This paper investigates the volume of unions of $ ext{delta}$ tubes in three-dimensional space under convex containment constraints, proving that Kakeya sets in $ ext{R}^3$ have full Minkowski and Hausdorff dimension.

Contribution

It establishes a new volume estimate for unions of convexly constrained tubes and confirms the Kakeya set conjecture in three dimensions.

Findings

Union of tubes has near-maximal volume under constraints

Kakeya sets in $ ext{R}^3$ have Minkowski dimension 3

Kakeya sets in $ ext{R}^3$ have Hausdorff dimension 3

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.