A generalization of Reifenberg's theorem in R 3
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-Hölder image of the unit ball in
R
m
. In this paper we prove that a subset of
R
3
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
R
n