Injective Convex Polyhedra
Ma\"el Pav\'on

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 -dimensional normed space is injective or equivalently is an absolute 1-Lipschitz retract if and only if is linearly isometric to (i.e., endowed with the -metric). We give an effective convex geometric characterization of injective convex polyhedra in . 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 -metric.
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