Injective Convex Polyhedra

We characterize the convex polyhedra P in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} for which any family of n-dimensional axis-parallel hypercubes centered in P and intersected with P has the binary intersection property. The characterization is effective, concrete and convex geometric. As an application, we prove that the convex polyhedra determined by a finite linear system of inequalities with at most two variables per inequality are of this type. This provides in particular new examples of injective (or equivalently hyperconvex) metric spaces.