Uniform Property (S)

Abstract

We introduce and investigate a quantitative version of Steinhaus' property$(S)$ for Banach spaces, called the \emph{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 $L_1(\mu)$ with atomless measure$\mu$, $$ U_{L_1(\mu)}(d;a)=\Big(\tfrac{4a}{2+d}\wedge 1\Big)d. $$ The class of spaces with uniform$(S)$ is stable under ultrapowers, Bochner-$L_1$ constructions, and contains all Gurari\u{\i} 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$\ell_1(\Gamma)$, $$ \|x\|_S=\big(\|x\|_1^2+\|x\|_2^2\big)^{1/2}, $$ which endows$\ell_1(\Gamma)$ and all its ultrapowers with uniform$(S)$. These results settle, inZFC, several open questions about the quantitative geometry of property$(S)$ posed by Kochanek and the second-named author.