Injective subsets of \(l_{\infty}(I)\) (Q2013990)

Recall that a metric space $(X,d)$ is said to be injective if, for every isometric embedding $i:A \to B$ of metric spaces $(A, d_{A})$, $(B, d_{B})$ and every 1-Lipschitz map $f:A \to X$, there exists a 1-Lipschitz map $\bar{f}:B \to X$ such that $f=\bar{f} \circ i$. \par The main result of the paper under review states an explicit characterization of all injective subsets of the model space $l_{\infty}(I)$ of all bounded real-valued functions defined on a non-empty set $I$, endowed with the supremum norm. This theorem furnishes a concrete expression for each single subset of the model spaces $l_{\infty}(I)$. \par The proof of this characterization is based on the proof of the characterization in the particular case where $I$ is finite obtained in [\textit{D. Descombes}, Spaces with convex geodesic bicombings. Dissertation ETH Zurich, No. 23109 (2015)]. Moreover, the proof is constructive and thus provides an algorithmic method to compute the Isbell injective hull of a finite set of points.