Benjamin Martin: Maneuvering through measure theory and Kakeya’s needle problem

Abstract.

The aim of this thesis is to study Kakeya’s needle problem, finding the smallest figure in the plane in which a unit segment can be rotated 180°, using the precision of measure theory. Measure theory is introduced abstractly and two measuring functions are defined, the Hausdorff measure and Lebesgue measure. The former extends the concept of dimension from the integers to the non-negative real numbers. Furthermore the equivalence between the Hausdorff measure and Lebesgue measure is shown. Finally the Kakeya conjecture, an open problem in mathematics, is briefly discussed with respect to the concepts introduced in this thesis.