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Published online by Cambridge University Press:  30 October 2015

Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA;
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA;


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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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Badger, M., ‘Harmonic polynomials and tangent measures of harmonic measure’, Rev. Mat. Iberoam. 27(3) (2011), 841870.CrossRefGoogle Scholar
Badger, M., ‘Flat points in zero sets of harmonic polynomials and harmonic measure from two sides’, J. Lond. Math. Soc. (2) 87(1) (2013), 111137.CrossRefGoogle Scholar
Beer, G., Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, 268 (Kluwer Academic Publishers Group, Dordrecht, 1993).CrossRefGoogle Scholar
Cheeger, J. and Colding, T. H., ‘On the structure of spaces with Ricci curvature bounded below. I’, J. Differential Geom. 46(3) (1997), 406480.CrossRefGoogle Scholar
David, G., ‘Hölder regularity of two-dimensional almost-minimal sets in ℝn’, Ann. Fac. Sci. Toulouse Math. (6) 18(1) (2009), 65246.CrossRefGoogle Scholar
David, G., De Pauw, T. and Toro, T., ‘A generalization of Reifenberg’s theorem in ℝ3’, Geom. Funct. Anal. 18(4) (2008), 11681235.CrossRefGoogle Scholar
David, G., Kenig, C. and Toro, T., ‘Asymptotically optimally doubling measures and Reifenberg flat sets with vanishing constant’, Comm. Pure Appl. Math. 54(4) (2001), 385449.3.0.CO;2-M>CrossRefGoogle Scholar
David, G. and Semmes, S., Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, 38 (American Mathematical Society, Providence, RI, 1993).CrossRefGoogle Scholar
David, G. and Semmes, S., ‘Fractured fractals and broken dreams’, inSelf-similar Geometry Through Metric and Measure, Oxford Lecture Series in Mathematics and its Applications, 7 (The Clarendon Press, Oxford University Press, New York, 1997).Google Scholar
David, G. and Toro, T., ‘Reifenberg parameterizations for sets with holes’, Mem. Amer. Math. Soc. 215(1012) (2012), vi+102.Google Scholar
Jones, P. W., ‘Rectifiable sets and the traveling salesman problem’, Invent. Math. 102(1) (1990), 115.CrossRefGoogle Scholar
Kenig, C., Preiss, D. and Toro, T., ‘Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions’, J. Amer. Math. Soc. 22(3) (2009), 771796.CrossRefGoogle Scholar
Kenig, C. and Toro, T., ‘Free boundary regularity below the continuous threshold: 2-phase problems’, J. Reine Angew. Math. 596 (2006), 144.CrossRefGoogle Scholar
Kenig, C. E. and Toro, T., ‘Harmonic measure on locally flat domains’, Duke Math. J. 87(3) (1997), 509551.CrossRefGoogle Scholar
Kenig, C. E. and Toro, T., ‘Free boundary regularity for harmonic measures and Poisson kernels’, Ann. of Math. (2) 150(2) (1999), 369454.CrossRefGoogle Scholar
Kirchheim, B. and Preiss, D., ‘Uniformly distributed measures in Euclidean spaces’, Math. Scand. 90(1) (2002), 152160.CrossRefGoogle Scholar
Kowalski, O. and Preiss, D., ‘Besicovitch-type properties of measures and submanifolds’, J. Reine Angew. Math. 379 (1987), 115151.Google Scholar
Le Donne, E., ‘Metric spaces with unique tangents’, Ann. Acad. Sci. Fenn. Math. 36(2) (2011), 683694.CrossRefGoogle Scholar
Le Donne, E., ‘A metric characeterization of Carnot groups’, Proc. Amer. Math. Soc. 143(2) (2015), 845849.CrossRefGoogle Scholar
Lewis, J. L. and Nyström, K., ‘Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains’, Adv. Calc. Var. 1(2) (2008), 133170.CrossRefGoogle Scholar
Lewis, S., ‘Singular points of Hölder asymptotically optimally doubling measures’, Calc. Var. Partial Differential Equations (2015), 47; doi:10.1007/s00526-015-0918-y.CrossRefGoogle Scholar
Liang, X., ‘Almgren-minimality of unions of two almost orthogonal planes in ℝ4’, Proc. Lond. Math. Soc. (3) 106(5) (2013), 10051059.CrossRefGoogle Scholar
Liang, X., ‘Almgren and topological minimality for the set Y× Y’, J. Funct. Anal. 266(10) (2014), 60076054.CrossRefGoogle Scholar
Luu, T. D., ‘On some properties of three-dimensional minimal sets in ℝ4’, Ann. Fac. Sci. Toulouse Math. (6) 22(3) (2013), 465493.CrossRefGoogle Scholar
Mattila, P. and Vuorinen, M., ‘Linear approximation property, Minkowski dimension, and quasiconformal spheres’, J. Lond. Math. Soc. (2) 42(2) (1990), 249266.CrossRefGoogle Scholar
Mattila, P., ‘Geometry of sets and measures in Euclidean spaces’, inFractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44 (Cambridge University Press, Cambridge, 1995).Google Scholar
Naber, A. and Valtorta, D., ‘Volume estimates on the critical sets of solutions to elliptic PDEs’, Preprint, 2014. arXiv:1403.4176.Google Scholar
Okikiolu, K., ‘Characterization of subsets of rectifiable curves in Rn’, J. Lond. Math. Soc. (2) 46(2) (1992), 336348.CrossRefGoogle Scholar
Preiss, D., Tolsa, X. and Toro, T., ‘On the smoothness of Hölder doubling measures’, Calc. Var. Partial Differential Equations 35(3) (2009), 339363.CrossRefGoogle Scholar
Preiss, D., ‘Geometry of measures in Rn: distribution, rectifiability, and densities’, Ann. of Math. (2) 125(3) (1987), 537643.CrossRefGoogle Scholar
Reifenberg, E. R., ‘Solution of the Plateau problem for m-dimensional surfaces of varying topological type’, Acta Math. 104 (1960), 192.CrossRefGoogle Scholar
Tyrrell Rockafellar, R. and Wets, R. J.-B., Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317 (Springer, Berlin, 1998).CrossRefGoogle Scholar
Taylor, J. E., ‘The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces’, Ann. of Math. (2) 103(3) (1976), 489539.CrossRefGoogle Scholar
Tolsa, X., ‘Uniform measures and uniform rectifiability’, J. Lond. Math. Soc. (2) 92(1) (2015), 118.CrossRefGoogle Scholar
Toro, T., ‘Geometric conditions and existence of bi-Lipschitz parameterizations’, Duke Math. J. 77(1) (1995), 193227.CrossRefGoogle Scholar
Walkup, D. W. and Wets, R. J.-B., ‘Continuity of some convex-cone-valued mappings’, Proc. Amer. Math. Soc. 18 (1967), 229235.CrossRefGoogle Scholar