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

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.