Sphere version of Banach--Kadec--Paley theorem
Abstract
We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical Lp-spaces. Our main result is a sphere version of the classical Banach--Kadec--Paley theorem: for 1≤ p,q<∞, the unit sphere SLq admits an almost Lipschitz embedding into Lp if and only if Lq admits a linear embedding into Lp. Consequently, the almost Lipschitz geometry of Lp-spheres completely determines the underlying linear structure of Lp-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton--Randrianarivony's concentration inequalities, Mendel--Naor's metric cotype inequality, and Naor's sharp metric Xp inequality. They also yield several sharp results on the Hölder geometry of Lp-spheres.
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BibTeX
@misc{cheng2026sphere,
title = {{Sphere version of Banach--Kadec--Paley theorem}},
author = {Qingjin Cheng and Wuyi He and Bo Xiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14641},
eprint = {2609.14641},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14641},
}
RIS
TY - UNPB
TI - Sphere version of Banach--Kadec--Paley theorem
AU - Qingjin Cheng
AU - Wuyi He
AU - Bo Xiang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.14641
UR - https://arxiv.org/abs/2609.14641
AB - We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical L_p-spaces. Our main result is a sphere version of the classical Banach--Kadec--Paley theorem: for 1≤ p,q<∞, the unit sphere S_L_q admits an almost Lipschitz embedding into L_p if and only if L_q admits a linear embedding into L_p. Consequently, the almost Lipschitz geometry of L_p-spheres completely determines the underlying linear structure of L_p-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton--Randrianarivony's concentration inequalities, Mendel--Naor's metric cotype inequality, and Naor's sharp metric X_p inequality. They also yield several sharp results on the Hölder geometry of L_p-spheres.
ER -