Uniform Property (S)

TL;DR

This paper introduces a quantitative version of Steinhaus' property (S) for Banach spaces, computes it for $L_1$ spaces, and shows its stability under various constructions, providing new insights into the geometry of Banach spaces.

Contribution

The paper defines the uniform property (S), computes it for $L_1$ spaces, and demonstrates its stability under ultrapowers, Bochner-$L_1$ constructions, and embeddings into spaces with this property.

Findings

Exact computation of the uniform (S) modulus for $L_1()$ spaces.

Stability of the uniform (S) property under ultrapowers and Banach space constructions.

Existence of explicit renormings of $\u2113_1()$ with uniform (S).

Abstract

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.