A submetric characterization of Rolewicz's property ($\beta$)

TL;DR

This paper provides a submetric characterization of Banach spaces with Rolewicz's property ($eta$), showing stability under coarse Lipschitz embeddings and quotients, advancing understanding of geometric properties in Banach space theory.

Contribution

It introduces a submetric characterization of Rolewicz's property ($eta$) and demonstrates its stability under coarse Lipschitz embeddings and quotients.

Findings

Characterization of Banach spaces with Rolewicz's property ($eta$)

Stability of property ($eta$) under coarse Lipschitz embeddings

Stability of property ($eta$) under coarse quotients

Abstract

The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ($\beta$). As applications we prove that up to renorming, property ($\beta$) is stable under coarse Lipschitz embeddings and coarse quotients.