Embeddability and rectifiability of Lipschitz differentiability spaces
Abstract
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 Lip−lip-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces (X,μ)⊂ℓ2 whose Gromov--Hausdorff tangents are Hausdorff limits of r−1 (X−x) in ℓ2 as r→ 0.
Related
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- Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas
- Lipschitz numbers and fine differentiability on metric measure spaces
- Differentiability of Lipschitz curves in p-Banach spaces
- Almost Homogeneity Characterizes Almost Bi-Lipschitz Euclidean Embeddability
BibTeX
@misc{caamano2026embeddability,
title = {{Embeddability and rectifiability of Lipschitz differentiability spaces}},
author = {Ivan Caamano and Sylvester Eriksson-Bique and Elefterios Soultanis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06700},
eprint = {2609.06700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Embeddability and rectifiability of Lipschitz differentiability spaces
AU - Ivan Caamano
AU - Sylvester Eriksson-Bique
AU - Elefterios Soultanis
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06700
UR - https://arxiv.org/abs/2609.06700
AB - 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 Lip−lip-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces (X,μ)⊂ℓ^2 whose Gromov--Hausdorff tangents are Hausdorff limits of r^−1 (X−x) in ℓ^2 as r→ 0.
ER -