Characterising rectifiable metric spaces using tangent spaces

TL;DR

This paper characterizes rectifiable subsets of complete metric spaces through local approximations by Euclidean spaces using tangent measures and Gromov--Hausdorff distance, extending classical tangent measure theory.

Contribution

It introduces a new tangent measure framework for metric spaces and provides a characterization of rectifiability via local Euclidean approximations.

Findings

Rectifiable sets are characterized by local Euclidean approximations.

A generalized tangent measure concept is developed for metric spaces.

The approach unifies classical tangent measures with Gromov--Hausdorff tangent spaces.

Abstract

We characterise rectifiable subsets of a complete metric space $X$ in terms of local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ with $\mathcal{H}^n(E)<\infty$ and has positive lower density almost everywhere, we prove that it is sufficient that, at almost every point and each sufficiently small scale, $E$ is approximated by a bi-Lipschitz image of Euclidean space. We also introduce a generalisation of Preiss's tangent measures that is suitable for the setting of arbitrary metric spaces and formulate our characterisation in terms of tangent measures. This definition is equivalent to that of Preiss when the ambient space is Euclidean, and equivalent to the measured Gromov--Hausdorff tangent space when the measure is doubling.