On asymptotically uniformly smoothness and nonlinear geometry of Banach spaces

TL;DR

This paper explores the nonlinear geometry of Banach spaces, focusing on asymptotic uniform smoothness, concentration inequalities, and embeddings, with implications for descriptive set theory and open problems.

Contribution

It introduces new results on concentration inequalities and embeddings in asymptotically uniformly smooth Banach spaces, and analyzes their descriptive set theoretical complexity.

Findings

Existence of concentration inequalities in asymptotically uniformly smooth spaces

Weakly sequentially continuous coarse embeddings studied

Descriptive set theoretical complexity of properties analyzed

Abstract

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.