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

Published online by Cambridge University Press:  30 October 2015

MATTHEW BADGER
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA; matthew.badger@uconn.edu
STEPHEN LEWIS
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA; iam@stephen-lewis.net

Abstract

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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 $n-4$.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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