A generalization of Reifenberg's theorem in R^N for flat cones

TL;DR

This paper extends Reifenberg's theorem by showing that sets close to cones over simplicial complexes are locally bi-Hölder equivalent to those cones, generalizing previous topological and geometric results.

Contribution

It generalizes Reifenberg's Topological Disk Theorem to cones over simplicial complexes, broadening the class of sets with controlled geometric structure.

Findings

Sets close to cones over simplicial complexes are locally bi-Hölder equivalent to those cones.

Generalizes Reifenberg's theorem from topological disks to more complex cone structures.

Extends results by David, De Pauw, and Toro from 2008.

Abstract

In this paper we prove that if a closed set in R^N is close to a cone over a simplicial complex at each point and at each scale, then it is locally bi-H\"older equivalent to such a cone. This generalizes Reifenberg's Topological Disk Theorem in 1960 and G. David, T. De Pauw and T. Toro's result in 2008.