Injective Convex Polyhedra

Abstract

It was shown by Nachbin in 1950 that an $n$-dimensional normed space $X$ is injective or equivalently is an absolute 1-Lipschitz retract if and only if $X$ is linearly isometric to $l_\infty^n$ (i.e., $\mathbb{R}^n$ endowed with the $l_{\infty}$-metric). We give an effective convex geometric characterization of injective convex polyhedra in $l_{\infty}^n$. As an application, we prove that if the set of solutions to a linear system of inequalities with at most two variables per inequality is non-empty, then it is injective when endowed with the $l_{\infty}$-metric.