PreprintCC BY-NC-ND 4.0arXiv
ε-isometries without isometric embeddings
Abstract
We show that there exist two separable real Banach spaces X and Y such that, for every ε>0, there is a standard exact ε-isometry fε: X→ Y whose distortion of every distance is between $0$ and ε, whereas no isometric embedding of X into Y exists, even without a linearity assumption. This gives a negative answer to \cite[Problem 1 and Problem 2]{cheng4} of Cheng and Zhou. The construction combines a snowflaked Lipschitz-free space with the Schur-property theorem of Kalton and the isometric linearization theorem of Godefroy and Kalton.
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BibTeX
@misc{sun2026εisometries,
title = {{$\varepsilon$-isometries without isometric embeddings}},
author = {Longfa Sun and Yipeng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13937},
eprint = {2609.13937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13937},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - ε-isometries without isometric embeddings
AU - Longfa Sun
AU - Yipeng Zhang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13937
UR - https://arxiv.org/abs/2609.13937
AB - We show that there exist two separable real Banach spaces X and Y such that, for every ε>0, there is a standard exact ε-isometry f_ε: X→ Y whose distortion of every distance is between $0$ and ε, whereas no isometric embedding of X into Y exists, even without a linearity assumption. This gives a negative answer to \cite[Problem 1 and Problem 2]{cheng4} of Cheng and Zhou. The construction combines a snowflaked Lipschitz-free space with the Schur-property theorem of Kalton and the isometric linearization theorem of Godefroy and Kalton.
ER -