Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spaces

The classical Reifenberg’s theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological Hölder manifold. Remarkably, this generalizes to metric spaces, where the approximation by planes is replaced by the Gromov-Hausdorff distance. This fact was shown by Cheeger and Colding (J Differ Geom 46:406–480, 1997) in an appendix of one of their celebrated works on Ricci limit spaces. Given the recent interest around this statement in the growing field of analysis in metric spaces, in this note we provide a self contained and detailed proof of the Cheeger and Colding result. Our presentation substantially expands the arguments in Cheeger and Colding (1997) and makes explicit all the relevant estimates and constructions. As a byproduct we also show a biLipschitz version of this result which, even if folklore among experts, was not present in the literature. This work is an extract from the doctoral dissertation of the second author.