Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
Abstract
Let X, Y be real Banach spaces with closed unit balls BX, BY and unit spheres SX, SY. We prove that every non-expansive map F: BX→ BY with F(SX)⊆ SY⊆ F(BX) is the restriction of a surjective linear isometry X→ Y. Consequently, a non-expansive bijection of BX onto itself that fails to be an isometry must map some point of SX into the open unit ball, so the plasticity of BX reduces to showing that non-expansive bijections preserve norm 1. The proof shows that F is Fréchet differentiable at $0$ with a surjective linear isometry T as its derivative, and then proves T−1F=id using a quantitative rigidity lemma for non-expansive self-maps H of BX satisfying |a+H(a)|=2 for a∈ SX; the latter relies on the Baillon-Bruck rate of asymptotic regularity for Krasnosel'skii-Mann iterations. As a corollary, the identity is extremal in the set of non-expansive self-maps of BX for every Banach space X, which together with a result of Bargetz, Dymond and Pirk shows that all surjective isometries of BX are extremal, answering a question of theirs, including the case of L1[0,1].
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BibTeX
@misc{kurik2026nonexpansive,
title = {{Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries}},
author = {Kaarel August Kurik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28111},
eprint = {2610.28111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28111},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
AU - Kaarel August Kurik
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28111
UR - https://arxiv.org/abs/2610.28111
AB - Let X, Y be real Banach spaces with closed unit balls B_X, B_Y and unit spheres S_X, S_Y. We prove that every non-expansive map F: B_X→ B_Y with F(S_X)⊆ S_Y⊆ F(B_X) is the restriction of a surjective linear isometry X→ Y. Consequently, a non-expansive bijection of B_X onto itself that fails to be an isometry must map some point of S_X into the open unit ball, so the plasticity of B_X reduces to showing that non-expansive bijections preserve norm 1. The proof shows that F is Fréchet differentiable at $0$ with a surjective linear isometry T as its derivative, and then proves T^−1F=id using a quantitative rigidity lemma for non-expansive self-maps H of B_X satisfying |a+H(a)|=2 for a∈ S_X; the latter relies on the Baillon-Bruck rate of asymptotic regularity for Krasnosel'skii-Mann iterations. As a corollary, the identity is extremal in the set of non-expansive self-maps of B_X for every Banach space X, which together with a result of Bargetz, Dymond and Pirk shows that all surjective isometries of B_X are extremal, answering a question of theirs, including the case of L^1[0,1].
ER -