LOCAL SET APPROXIMATION: MATTILA–VUORINEN TYPE SETS, REIFENBERG TYPE SETS, AND TANGENT SETS

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 develope...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2015-10, Vol.3, Article e24
Main Authors: BADGER, MATTHEW, LEWIS, STEPHEN
Format: Article
Language:English
Subjects:
28A75
35R35
49J52 (primary)
49Q20 (secondary)
Analysis
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Summary: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 $\mathbb{R}^{n}$ ($n\geqslant 4$) has upper Minkowski dimension at most $n-4$.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2015.26