Minimal half-spaces and external representation of tropical polyhedra

TL;DR

This paper characterizes minimal tropical half-spaces containing a tropical polyhedron, reveals their potential infinitude, disproves a related conjecture, and establishes an analogue of the Minkowski-Weyl theorem for tropical polyhedra.

Contribution

It provides a new characterization of minimal tropical half-spaces, shows their possible infinitude, and proves a tropical Minkowski-Weyl theorem linking internal and external representations.

Findings

Counterexample showing infinite minimal half-spaces

Disproof of a conjecture by Block and Yu

External representation via extreme elements of the tropical polar

Abstract

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.