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The renorming-stable ball covering property and C(K)-spaces

  • Rui Liu
  • Jie Shen
  • Richard J. Smith
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Abstract

A Banach space has the ball-covering property (BCP) if its unit sphere can be covered by countably many balls that miss the origin. A Banach space has the renorming-stable ball-covering property (RBCP) if it has the BCP with respect to every equivalent norm. We investigate the RBCP from the perspective of C(K)-spaces. In particular, we show that there are many non-separable C(K)-spaces having the RBCP. This yields an emphatic negative answer to a question of Cheng, Kato and Zhang from 2020.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026renormingstable,
  title = {{The renorming-stable ball covering property and $C(K)$-spaces}},
  author = {Rui Liu and Jie Shen and Richard J. Smith},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05846},
  eprint = {2610.05846},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05846},
}
RIS
TY  - UNPB
TI  - The renorming-stable ball covering property and C(K)-spaces
AU  - Rui Liu
AU  - Jie Shen
AU  - Richard J. Smith
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05846
UR  - https://arxiv.org/abs/2610.05846
AB  - A Banach space has the ball-covering property (BCP) if its unit sphere can be covered by countably many balls that miss the origin. A Banach space has the renorming-stable ball-covering property (RBCP) if it has the BCP with respect to every equivalent norm. We investigate the RBCP from the perspective of C(K)-spaces. In particular, we show that there are many non-separable C(K)-spaces having the RBCP. This yields an emphatic negative answer to a question of Cheng, Kato and Zhang from 2020.
ER  -