Local Set Approximation: Infinitesimal to Local Theorems for Sets in Euclidean Space and Applications

Abstract

In this thesis we develop the theory of Local Set Approximation (LSA), a framework which arises naturally from the study of sets with singularities. That is, we describe the local structure of a set A in Euclidean space through studying a class of sets S which approximates A well in small balls. We will give two interpretations LSA in Chapters 2 and 3. If in small balls B(x; r), our set A is close to some nice model set S, the approximation is unilateral. On the other hand, if in small balls B(x; r), our set A is close to S and S is close to A, the approximation is bilateral. Both of these models appear naturally in areas of geometric measure theory such as area minimizers, mass minimizers, free boundary, and regularity of measures. In Chapters 4 and 5, we give applications of local set approximation to the study of asymptotically optimally doubling measures.