Uniform Property (S)

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 $L_1(\mu)$ 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, in ZFC, several open questions about the quantitative geometry of property $(S)$ posed by Kochanek and the second-named author.