Geometric measure theory and Kakeya’s problem

Abstract

The main goal of this work is to define a 2-dimensional Besicovitch set and introduce the Kakeya problem, which asks for the minimal area of a set in which a segment can be rotated continuosly in the plane and can lay in all directions, approaching it both from a geometrical point of view and by using techniques of polar reciprocity. We first establish basic results about measures that we use throughout the work, paying special attention to Hausdorff’s measure and it’s properties. We then construct a Besicovitch set and a solution to Kakeya’s problem, first geometrically and then by duality and projection arguments before looking into the Hausdorff dimension of such sets.