Reifenberg Theorem for Locally Finitely Almost Splitting Sets

Abstract The well-known Reifenberg theorem states that if a subset of $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> 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 $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> which can be approximated by at most N parallel k planes at each point and scale. As a subset of $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> 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 $$\mathbb {R}^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> </mml:math> , which is itself a biHölder homeomorphism of the disk into the set of subsets of $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> .