Reifenberg Theorem for Locally Finitely Almost Splitting Sets

2025/08/20 by Zang, Jiaqi
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.14805

Abstract

The well-known Reifenberg theorem states that if a subset of ℝⁿ can be well approximated by k-planes at every point and every scale, then it is biHölder homeomorphic to a k-disk. This article concerns a subset S of ℝⁿ which can be approximated by at most N parallel k planes at each point and scale. As a subset of ℝⁿ such an S may be quite degenerate; S may clearly not be homeomorphic to a disk, and indeed we will see may not be homeomorphic to a union of disks. However, we prove that S is still the image of a multivalued map on ℝ^k, which is itself a biHölder homeomorphism of the disk into the set of subsets of ℝⁿ.

Citations

• Reifenberg Parameterizations for Sets with Holes

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