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Abstract: We give a characterization of the minimal tropical half-spaces containing agiven tropical polyhedron, from which we derive a counter example showing thatthe number of such minimal half-spaces can be infinite, contradicting somestatements which appeared in the tropical literature, and disproving aconjecture of F. Block and J. Yu. We also establish an analogue of theMinkowski-Weyl theorem, showing that a tropical polyhedron can be equivalentlyrepresented internally in terms of extreme points and rays or externally interms of half-spaces containing it. A canonical external representation of apolyhedron turns out to be provided by the extreme elements of its tropicalpolar. We characterize these extreme elements, showing in particular that theyare determined by support vectors.



Author: Stephane Gaubert, Ricardo D. Katz

Source: https://arxiv.org/



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