Injective Convex Polyhedra

TL;DR

This paper characterizes injective convex polyhedra in finite-dimensional $l_ abla$-spaces and shows certain linear inequality solution sets are injective under the $l_ abla$-metric, extending classical results on injective spaces.

Contribution

It provides a convex geometric characterization of injective convex polyhedra in $l_ abla^n$ and applies this to linear systems with limited variables per inequality.

Findings

Convex geometric characterization of injective polyhedra in $l_ abla^n$.

Linear inequality systems with at most two variables per inequality are injective.

Extension of classical injectivity results to specific convex polyhedra and linear systems.

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.