On Three Geometric Problems in Analysis

Chapter 1 concerns the construction of pathological sets of lines in R^n. Some examples include Besicovitch sets, which have zero Lebesgue measure and contain a line in every direction, and Davies sets, which are collections of lines that cover a set of positive Lebesgue measure while having zero measure outside of it. Our original contribution is the construction of 2-purely unrectifiable Besicovitch and Davies sets that consist of mutually disjoint lines. We also conside parametrizations of such sets that are Holder regular, providing examples that are optimal in this sense. We use a Baire Category approach. Chapter 2 concerns the construction of unrectifiable hypersurfaces in Carnot groups. We construct intrinsically C^{1, alpha} hypersurfaces, for every alpha between 0 and 1, that have fractional Euclidean dimension. Our construction is sharp in step-2 groups, as it attains the maximal Euclidean dimension compatible with the prescribed intrinsic regularity. Moreover, by slicing these hypersurfaces, we obtain lower-codimensional surfaces that remain Euclidean non-rectifiable. Chapter 3 focuses on the fractional isoperimetric inequality between the s-perimeter and the t-perimeter, for 0<s<t<1. We show that the constant provided by the ball is locally optimal: any sufficiently small graph perturbation of the ball leads to a strictly larger constant.