Effective Reifenberg theorems in Hilbert and Banach spaces

Abstract

The aim of this article is to study effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from ${\mathbb{R}}^n$ continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a weak version of the effective Reifenberg theorem holds in that if one assumes a summability estimate ${\int }_0^2{\beta }^k(x,r{)}^1\frac{dr}{r}<M$ {\it without power gain}, then $\mu$ must again be rectifiable with measure estimates. Improving this estimate in order to obtain a power gain turns out to be a subtle issue. For $k=1$ we will see for a {\it uniformly smooth} Banach space that if ${\int }_0^2{\beta }^1(x,r{)}^{\alpha }\frac{dr}{r}<M^{\alpha /2}$, where $\alpha$ is the smoothness power of the Banach space, then $\mu$ is again rectifiable with uniform measure estimates. %We will provide examples showing that this power gain is sharp, and that for $k>1$ any power gain at all may fail, even for uniformly smooth Banach spaces.