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- W4287323237 abstract "In this note we compare two ways of measuring the $n$-dimensional flatness of a set $Ssubset mathbb{R}^d$, where $nin mathbb{N}$ 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)." @default.
- W4287323237 created "2022-07-25" @default.
- W4287323237 creator A5078048848 @default.
- W4287323237 date "2021-02-25" @default.
- W4287323237 modified "2023-09-25" @default.
- W4287323237 title "A remark on two notions of flatness for sets in the Euclidean space" @default.
- W4287323237 doi "https://doi.org/10.48550/arxiv.2102.12910" @default.
- W4287323237 hasPublicationYear "2021" @default.
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