The Kakeya conjecture in two dimensions

Abstract

We investigate the proof of the Kakeya conjecture in two dimensions, which states that every set which contains a unit line segment in every direction of the plane must have Hausdorff dimension of 2. In this case, we look at a theorem which states that every subset of the plane which contains a line in every direction must have Hausdorff dimension of at least 2.

We also construct a set with a line segment in every direction which has 2-dimensional Hausdorff measure 0.