On asymptotic B-convexity and infratype

TL;DR

This paper introduces the concepts of asymptotic B-convexity and infratype p, providing simplified proofs of asymptotic versions of classical theorems in Banach space theory, and discusses asymptotic stable type.

Contribution

It defines new asymptotic geometric properties and proves their fundamental theorems, extending classical results to the asymptotic setting with simplified proofs.

Findings

Established asymptotic B-convexity and infratype p.

Proved asymptotic analogs of Giesy and Pisier's results.

Provided a simplified proof of an asymptotic version of Pisier's $\, ext{l}_1$-theorem.

Abstract

In this note, we introduce and study the notions of asymptotic B-convexity and asymptotic infratype $p$, and we prove asymptotic analogs of a series of results due to Giesy \cite{Giesy66} and Pisier \cite{Pisier74}. In particular, we give a simplified proof of an asymptotic version of Pisier's $\ell_1$-theorem that was originally proven by Causey, Draga, and Kochanek in \cite{CDK19}. We also briefly discuss the notion of asymptotic stable type.