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 $$\underset{\Vert z\Vert \le a}{\sup }|\Vert x+z\Vert -\Vert y+z\Vert |$$ is bounded below by a positive function of$a$ and$\Vert x-y\Vert$. We compute this modulus exactly for the spaces $L_1(\hat{I}¼)$ with atomless measure$\hat{I}¼$, $$U_{L_1(\hat{I}¼)}(d;a)=(\frac{4a}{2+d}\wedge 1)d.$$ 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${\ell }_1(\hat{I}“)$, $$\Vert x{\Vert }_S=(\Vert x{\Vert }_1^2+\Vert x{\Vert }_2^2{)}^{1/2},$$ which endows${\ell }_1(\hat{I}“)$ 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.

22 pp, accepted for publication in Studia Mathematica