A Generalization of Reifenberg's Theorem in R-3

David, Guy;De Pauw, Thierry;Toro, Tatiana
(2008) Geometric and functional analysis — Vol. 18, n° 4, p. 1168-1235 (2008)

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Authors
  • David, Guy
    Author
  • De Pauw, ThierryUCLouvain
    Author
  • Toro, Tatiana
    Author
Abstract
In 1960 Reifenberg proved the topological disc property. He showed that a subset of R-n which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-Holder image of the unit ball in R-m. In this paper we prove that a subset of R-3 which is well approximated in the Hausdorff distance sense by one of the three standard area-minimizing cones at each point and at each (small) scale is locally a bi-Holder deformation of a minimal cone. We also prove an analogous result for more general cones in R-n.
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David, G., De Pauw, T., & Toro, T. (2008). A Generalization of Reifenberg’s Theorem in R-3. Geometric and functional analysis, 18(4), 1168-1235. https://doi.org/10.1007/s00039-008-0681-8 (Original work published 2008)