Geodesic
LOCAL SET APPROXIMATION: MATTILA–VUORINEN TYPE SETS, REIFENBERG TYPE SETS, AND TANGENT SETS
Forum of Mathematics, Sigma, Tome 3 (2015)

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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 $n-4$. 
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     author = {MATTHEW BADGER and STEPHEN LEWIS},
     title = {LOCAL {SET} {APPROXIMATION:} {MATTILA{\textendash}VUORINEN} {TYPE} {SETS,} {REIFENBERG} {TYPE} {SETS,} {AND} {TANGENT} {SETS}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2015},
     doi = {10.1017/fms.2015.26},
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MATTHEW BADGER; STEPHEN LEWIS. LOCAL SET APPROXIMATION: MATTILA–VUORINEN TYPE SETS, REIFENBERG TYPE SETS, AND TANGENT SETS. Forum of Mathematics, Sigma, Tome 3 (2015). doi: 10.1017/fms.2015.26

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