A generalization of Reifenberg's theorem in R³

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In 1960 Reifenberg proved the topological disc property. He showed that a subset of Rⁿ which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-Hölder image of the unit ball in Rᵐ. In this paper we prove that a subset of R³ which is well approximated by a minimal cone at each point and at each (small) scale is locally a bi-Hölder deformation of a minimal cone. We also prove an analogous result for more general cones in Rⁿ