Effective Reifenberg theorems in Hilbert and Banach spaces

A famous theorem by Reifenberg states that closed subsets of Rn that look sufficiently close to k-dimensional at all scales are actually C, γ equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure μ in Rn, one may introduce the k-dimensional Jones’ β-numbers of the measure, where βμk(x,r) quantifies on a given ball Br(x) how closely in an integral sense the support of the measure is to living inside a k-dimensional subspace. Recently, it has been proven that if these β-numbers satisfy the uniform summability estimate ∫02βμk(x,r)2drr