王虹：凸集的并集体积估计，以及三维中Kakeya集合猜想（英文版）

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 .