Besicovitch sets and regularity

This work prove the counterintuitive existence of a set with 'area' zero which contains for every direction a segment of length 1 parallel to that direction. This also proves that in the plane is not true that, given a set with finite measure, for almost every direction the function that associate to the distance of a plane orthogonal to that direction, the measure of the section of the set cut by that plane, is continuous . However this is true in higher dimension as it is proved by this work.