Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets

TL;DR

This paper develops a unified framework for local set approximation in Euclidean spaces, extending classical geometric measure theory concepts and applying them to problems involving the structure and dimension of singular sets.

Contribution

It introduces a comprehensive approach to local set approximation that generalizes previous theories and demonstrates applications to variational problems and measure-theoretic singularities.

Findings

The singular part of certain measures has upper Minkowski dimension at most n-4.

Unified framework extends and connects ideas of Jones, Reifenberg, and Preiss.

Applications to geometric measure theory and PDEs.

Abstract

We investigate the interplay between the local and asymptotic geometry of a set $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S} \subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^n$ ($n\geq 4$) has upper Minkowski dimension at most $n-4$.