Hamming graphs and concentration properties in non-quasi-reflexive Banach spaces

TL;DR

This paper investigates concentration properties of Lipschitz maps on Hamming graphs within non-quasi-reflexive Banach spaces, extending known results and constructing new examples of spaces with specific concentration inequalities.

Contribution

It extends Causey's result to $\, ext{l}_p$-sums of spaces, providing the first examples of non-quasi-reflexive Banach spaces with concentration inequalities.

Findings

Extended coarse Lipschitz structure results to $\, ext{l}_p$-sums.

Constructed examples of non-quasi-reflexive spaces with concentration properties.

Provided conditions for a space to be asymptotic-$c_0$ based on concentration.

Abstract

In this note, we study some concentration properties for Lipschitz maps defined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of quasi-reflexive spaces satisfying upper $\ell_p$ tree estimates to the setting of $\ell_p$-sums of such spaces. Our result provides us with a tool for constructing the first examples of Banach spaces that are not quasi-reflexive but nevertheless admit some concentration inequality. We also give a sufficient condition for a space to be asymptotic-$c_0$ in terms of a concentration property, as well as relevant counterexamples.