On coarse Lipschitz embeddability into $c_0(\kappa)$

TL;DR

This paper characterizes coarse Lipschitz embeddability of metric spaces into $c_0( appa)$, extending Pelant's 1994 uniform embeddability characterization, and shows equivalence of various embeddability notions for normed spaces.

Contribution

It provides a new characterization of coarse Lipschitz embeddability into $c_0( appa)$ and establishes the equivalence of different embeddability concepts for normed linear spaces.

Findings

Coarse Lipschitz embeddability characterized similarly to Pelant's uniform embeddability.

Coarse, uniform, and bi-Lipschitz embeddability are equivalent for normed spaces.

Extension of embedding characterizations to broader classes of metric spaces.

Abstract

In 1994, Jan Pelant proved that a metric property related to the notion of paracompactness called the uniform Stone property characterizes a metric space's uniform embeddability into $c_0(\kappa)$ for some cardinality $\kappa$. In this paper it is shown that coarse Lipschitz embeddability of a metric space into $c_0(\kappa)$ can be characterized in a similar manner. It is also shown that coarse, uniform, and bi-Lipschitz embeddability into $c_0(\kappa)$ are equivalent notions for normed linear spaces.