Asymptotic geometry of Banach spaces and uniform quotient maps

TL;DR

This paper explores the relationship between the asymptotic geometry of Banach spaces and uniform quotient maps, establishing how the smoothness modulus of the range relates to the property $(eta)$ of the domain.

Contribution

It demonstrates that the modulus of asymptotic uniform smoothness of the range space can be compared to the property $(eta)$ of the domain space, with conditions for improvement.

Findings

The modulus of asymptotic uniform smoothness of the range space can be bounded by the property $(eta)$ of the domain.

Conditions are provided under which this comparison is enhanced.

The work advances understanding of geometric properties in nonlinear quotient problems.

Abstract

Recently, Lima and Randrianarivony pointed out the role of the property $(\beta)$ of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of $(\beta)$ of the domain space. We also provide conditions under which this comparison can be improved.