Sperner theory
Methods and Applications of (MAX, +) Linear Algebra
STACS '97 Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science
Tropical convexity via cellular resolutions
Journal of Algebraic Combinatorics: An International Journal
Carathéodory, Helly and the Others in the Max-Plus World
Discrete & Computational Geometry
The number of extreme points of tropical polyhedra
Journal of Combinatorial Theory Series A
RelMiCS'06/AKA'06 Proceedings of the 9th international conference on Relational Methods in Computer Science, and 4th international conference on Applications of Kleene Algebra
Minimal external representations of tropical polyhedra
Journal of Combinatorial Theory Series A
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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.