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

We study sets of $δ$ 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.