Connections between metric characterizations of superreflexivity and the Radon-Nikod\'ym property for dual Banach spaces

TL;DR

This paper explores the relationship between metric characterizations of superreflexivity and the Radon-Nikodým property in dual Banach spaces, revealing that finite subsets characterizing RNP also characterize superreflexivity, but not vice versa.

Contribution

It establishes that finite metric subsets characterizing RNP for dual spaces also characterize superreflexivity, and discusses limitations of the converse implication.

Findings

Finite subsets characterizing RNP also characterize superreflexivity.

The converse does not hold;  is a counterexample.

The main result links metric characterizations of RNP and superreflexivity.

Abstract

Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikod\'ym property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample.