Cheng–Kato–Zhang question on the renorming-stable ball-covering property

Is every Banach space $X$ with the renorming-stable ball-covering property separable? More precisely, if for every norm $\lVert\cdot\rVert'$ equivalent to the given norm on $X$ , the unit sphere $S_{(X,\lVert\cdot\rVert')}$ can be covered by countably many balls $\overline{B}_{\lVert\cdot\rVert'}(x_n,r_n)$ satisfying $0\notin\overline{B}_{\lVert\cdot\rVert'}(x_n,r_n)$ , must $X$ be separable?

References

Primary source

arXiv

Additional references

The renorming-stable ball covering property and C(K)-spaces — arXiv — Rui Liu, Jie Shen, Richard J. Smith

banach-spaces functional-analysis

Posted 2 days ago · edited 2 days ago

Progress summary

Refreshed 2 days ago

Claimed solved

An October 2026 preprint claims that even nonseparable spaces of continuous functions can have the ball-covering property under every equivalent norm, answering the 2020 question negatively.

The 2020 question asks whether nonseparability rules out this renorming-stable property. The new claim says it does not, via examples among spaces of continuous functions.

October 2026 preprint

Rui Liu, Jie Shen, and Richard J. Smith claim that nonseparable $C(K)$ -spaces can have the ball-covering property for every equivalent norm. If correct, this gives a negative answer to the Cheng–Kato–Zhang question, but the retrieved evidence is an unrefereed preprint without independent mathematical assessment.

Current status (as of October 2026): The question is claimed answered negatively by a preprint, but the claim remains unverified; no independently assessed resolution was found.

Sources

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