preprint

Inserted: 21 jun 2024
Last Updated: 21 jun 2024

Year: 2024

Abstract:

The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological H\"older 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 in an appendix of one of their celebrated works on Ricci limit spaces 8. 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 8 and makes explicit all the relevant estimates and constructions. As a byproduct we also shows 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.