A remark on two notions of flatness for sets in the Euclidean space

TL;DR

This paper compares two notions of measuring the flatness of sets in Euclidean space, revealing a quadratic relationship between them and applying this insight to extend a fundamental geometric theorem.

Contribution

It establishes a rigorous connection between classical and intrinsic flatness measures, showing the intrinsic measure behaves as the square of the classical one, and extends the Cheeger-Colding theorem.

Findings

The intrinsic flatness measure ${ m a}(x,r)$ behaves as the square of the classical Reifenberg-flat number $eta(x,r)$.

The main result shows ${ m a}(x,r)$ is comparable to $ig(eta(x,r)ig)^2$.

Application to extending the Cheeger-Colding intrinsic-Reifenberg theorem to biLipschitz cases.

Abstract

In this note we compare two ways of measuring the $n$-dimensional "flatness" of a set $S\subset \mathbb{R}^d$, where $n\in \mathbb{N}$ and $d>n$. The first one is to consider the classical Reifenberg-flat numbers $\alpha(x,r)$ ($x \in S$, $r>0$), which measure the minimal scaling-invariant Hausdorff distances in $B_r(x)$ between $S$ and $n$-dimensional affine subspaces of $\mathbb{R}^d$. The second is an `intrinsic' approach in which we view the same set $S$ as a metric space (endowed with the induced Euclidean distance). Then we consider numbers ${\sf a}(x,r)$'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at $x$ of radius $r$ in $S$ and the $n$-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers ${\sf a}(x,r)$'s behaves as the square of the numbers $\alpha(x,r)$'s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones' numbers $\beta$'s (i.e. the one-sided version of the numbers $\alpha$'s).