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Uniform Property (S)

William B. Johnson, Tomasz Kania

TL;DR

This work introduces a quantitative version of Steinhaus' property (S), called uniform property (S), by defining the modulus $U_X(x,y;a)$ and its uniform lower bound $U_X(d;a)$. The authors derive the exact $L_1$-case formula $U_{L_1(μ)}(d;a)=\Big(\frac{4a}{2+d} \wedge 1\Big) d$ for atomless measures, establish stability under ultrapowers and Bochner-L1 constructions, and show that Gurariĭ spaces and Banach lattices of almost universal disposition possess uniform (S); they also prove every Banach space embeds isometrically into a non-strictly convex space with uniform (S) and construct an explicit equivalent renorming of $\ell_1(Γ)$, $\|x\|_S=(\|x\|_1^2+\|x\|_2^2)^{1/2}$, which endows $\ell_1(Γ)$ and its ultrapowers with uniform (S). These results resolve several open questions on the quantitative geometry of (S) in ZFC and reveal a robust landscape where uniform (S) is preserved under common constructions and renormings.

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 $L_1(μ)$ 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 Gurariĭ 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.

Uniform Property (S)

TL;DR

This work introduces a quantitative version of Steinhaus' property (S), called uniform property (S), by defining the modulus and its uniform lower bound . The authors derive the exact -case formula for atomless measures, establish stability under ultrapowers and Bochner-L1 constructions, and show that Gurariĭ spaces and Banach lattices of almost universal disposition possess uniform (S); they also prove every Banach space embeds isometrically into a non-strictly convex space with uniform (S) and construct an explicit equivalent renorming of , , which endows and its ultrapowers with uniform (S). These results resolve several open questions on the quantitative geometry of (S) in ZFC and reveal a robust landscape where uniform (S) is preserved under common constructions and renormings.

Abstract

We introduce and investigate a quantitative version of Steinhaus' property for Banach spaces, called the uniform property . A Banach space is said to have uniform if for every pair of distinct unit vectors and every , the difference of the perturbed norms is bounded below by a positive function of and . We compute this modulus exactly for the spaces with atomless measure , The class of spaces with uniform is stable under ultrapowers, Bochner- constructions, and contains all Gurariĭ 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 . We further exhibit an explicit equivalent renorming of , which endows and all its ultrapowers with uniform . These results settle, in ZFC, several open questions about the quantitative geometry of property posed by Kochanek and the second-named author.
Paper Structure (8 sections, 14 theorems, 93 equations)

This paper contains 8 sections, 14 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs
  4. Proof of Theorem B${}^\prime$
  5. Proof of Theorem \ref{['ThmC']}
  6. Proof of Theorem \ref{['thm:l1renorming']}
  7. Appendix: Banach lattices of almost universal disposition for finitely generated Banach lattices
  8. Open problems

Key Result

Lemma 1.3

Fix $p>1$ and $a\in(0,1)$. If $r,s\in[1-a,1+a]$, then by the mean value theorem Consequently, for every normed space $X$ and every $a\in(0,1)$ we have, for $0<d<2$, In particular, for any $a\in(0,1)$ and $p>1$, $U_X^{(p)}(d;a)>0$ if and only if $U_X(d;a)>0$.

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Theorem 1
  • Corollary 1.6
  • Corollary 1.7
  • ...and 21 more