On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson
Abstract
Let X be a normed space and SX be the unit sphere of X. We say that X has Property (H) if there exist two increasing sequences Vj⊂ X and Hj⊂ ℓ2, j=1,2,…, of finite dimensional subspaces such that V=⋃j Vj is dense in X, and there is a uniformly continuous mapping ψ:SV→ Sℓ2 such that the restrictions ψ|SVj, j=1,2,…, are isomorphisms from SVj onto SHj. The Kasparov-Yu problem asks whether the Banach space c0 of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that c0 admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum μn of the Lipschitz constants of all nonzero-degree maps from Sℓ∞n to Sℓ2n satisfies limn→∞ μn/log n=1/2. (2) Let λn and Cn denote the infima of the Lipschitz constants of homeomorphisms from Sℓ∞n onto Sℓ2n and from Bℓ∞n onto Bℓ2n, respectively. Then limn→∞ λn/log n=limn→∞ Cn/log n=1/2. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.
Related
BibTeX
@misc{cheng2026property,
title = {{On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson}},
author = {Lixin Cheng and Qingjin Cheng and Yue Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15817},
eprint = {2609.15817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15817},
}
RIS
TY - UNPB
TI - On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson
AU - Lixin Cheng
AU - Qingjin Cheng
AU - Yue Wang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.15817
UR - https://arxiv.org/abs/2609.15817
AB - Let X be a normed space and S_X be the unit sphere of X. We say that X has Property (H) if there exist two increasing sequences V_j⊂ X and H_j⊂ ℓ_2, j=1,2,…, of finite dimensional subspaces such that V=⋃_j V_j is dense in X, and there is a uniformly continuous mapping ψ:S_V→ S_ℓ_2 such that the restrictions ψ|_S_V_j, j=1,2,…, are isomorphisms from S_V_j onto S_H_j. The Kasparov-Yu problem asks whether the Banach space c_0 of all null sequences admits Property (H), or rational Property (H). In this paper, we prove that c_0 admits neither Property (H) nor rational Property (H). We also obtain the following results. (1) The infimum μ_n of the Lipschitz constants of all nonzero-degree maps from S_ℓ_∞^n to S_ℓ_2^n satisfies lim_n→∞ μ_n/log n=1/2. (2) Let λ_n and C_n denote the infima of the Lipschitz constants of homeomorphisms from S_ℓ_∞^n onto S_ℓ_2^n and from B_ℓ_∞^n onto B_ℓ_2^n, respectively. Then lim_n→∞ λ_n/log n=lim_n→∞ C_n/log n=1/2. The first result solves a problem of Gromov on nonzero-degree maps between finite-dimensional spheres, while the second settles Johnson's problem on the Lipschitz distortion of finite-dimensional unit balls.
ER -