Embeddability and rectifiability of Lipschitz differentiability spaces
We prove that Lipschitz differentiability spaces which bi-Lipschitz embed into an RNP-space are countably rectifiable. In contrast to earlier methods of Cheeger and Kleiner, our approach does not rely on differentiating RNP-targets, and uses instead decomposability bundles and a careful blow-up analysis. We also present decomposability bundles in a way which avoids the mention of Alberti representations and generalizes the approach of Alberti--Marchese to measures in RNP-spaces. We moreover study fragment-wise differentiability into RNP-targets, give a new ${\rm Lip}-{\rm lip}$-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces $(X,\mu)\subset\ell^2$ whose Gromov--Hausdorff tangents are Hausdorff limits of $r^{-1} (X-x)$ in $\ell^2$ as $r\to 0$.