Uniform Property (S)
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917261126664192 |
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| author | Johnson, William B. Kania, Tomasz |
| author_facet | Johnson, William B. Kania, Tomasz |
| contents | We introduce and investigate a quantitative version of Steinhaus' property $(S)$ for Banach spaces, called the uniform property $(S)$. A Banach space $X$ is said to have uniform $(S)$ if for every pair of distinct unit vectors $x,y\in X$ and every $a>0$, the difference of the perturbed norms \[
\sup_{\|z\|\le a}\big|\|x+z\|-\|y+z\|\big| \] is bounded below by a positive function of $a$ and $\|x-y\|$. We compute this modulus exactly for the spaces $L_1(μ)$ with atomless measure $μ$, \[
U_{L_1(μ)}(d;a)=\Big(\tfrac{4a}{2+d}\wedge 1\Big)d, \] The class of spaces with uniform $(S)$ is stable under ultrapowers, Bochner-$L_1$ constructions, and contains all Gurari\uı spaces as well as Banach lattices of almost universal disposition. In particular, every Banach space embeds isometrically into a non-strictly convex Banach space of the same density having uniform $(S)$. We further exhibit an explicit equivalent renorming of $\ell_1(Γ)$, \[
\|x\|_S=\big(\|x\|_1^2+\|x\|_2^2\big)^{1/2}, \] which endows $\ell_1(Γ)$ and all its ultrapowers with uniform $(S)$. These results settle, in ZFC, several open questions about the quantitative geometry of property $(S)$ posed by Kochanek and the second-named author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09106 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform Property (S) Johnson, William B. Kania, Tomasz Functional Analysis 46B04, 46B20 (primary), 11H06 (secondary) We introduce and investigate a quantitative version of Steinhaus' property $(S)$ for Banach spaces, called the uniform property $(S)$. A Banach space $X$ is said to have uniform $(S)$ if for every pair of distinct unit vectors $x,y\in X$ and every $a>0$, the difference of the perturbed norms \[ \sup_{\|z\|\le a}\big|\|x+z\|-\|y+z\|\big| \] is bounded below by a positive function of $a$ and $\|x-y\|$. We compute this modulus exactly for the spaces $L_1(μ)$ with atomless measure $μ$, \[ U_{L_1(μ)}(d;a)=\Big(\tfrac{4a}{2+d}\wedge 1\Big)d, \] The class of spaces with uniform $(S)$ is stable under ultrapowers, Bochner-$L_1$ constructions, and contains all Gurari\uı spaces as well as Banach lattices of almost universal disposition. In particular, every Banach space embeds isometrically into a non-strictly convex Banach space of the same density having uniform $(S)$. We further exhibit an explicit equivalent renorming of $\ell_1(Γ)$, \[ \|x\|_S=\big(\|x\|_1^2+\|x\|_2^2\big)^{1/2}, \] which endows $\ell_1(Γ)$ and all its ultrapowers with uniform $(S)$. These results settle, in ZFC, several open questions about the quantitative geometry of property $(S)$ posed by Kochanek and the second-named author. |
| title | Uniform Property (S) |
| topic | Functional Analysis 46B04, 46B20 (primary), 11H06 (secondary) |
| url | https://arxiv.org/abs/2602.09106 |