Injective Subsets of $l_{\infty}(I)$

TL;DR

This paper characterizes all injective subsets of the space l_infinity(I) using inequalities involving 1-Lipschitz functions, providing a comprehensive description of absolute 1-Lipschitz retracts within this space.

Contribution

It offers an explicit characterization of injective subsets of l_infinity(I) for any set I, linking them to inequalities with 1-Lipschitz functions, thus advancing understanding of Lipschitz retracts.

Findings

Characterization of injective subsets via inequalities

Identification of absolute 1-Lipschitz retracts in l_infinity(I)

Generalization to arbitrary index sets I

Abstract

We give an explicit characterization of all injective subsets of the model space $l_{\infty}(I)$ for a general set $I$, in terms of inequalities involving $1$-Lipschitz functions. Since the class of all injective metric spaces coincides with the one of all absolute $1$-Lipschitz retracts, the present work yields a characterization of all the subsets of $l_{\infty}(I)$ that are absolute $1$-Lipschitz retracts.