CONICET | Buscador de Institutos y Recursos Humanos

Tropical polyhedra are known to be representable externally, as intersections of finitely many tropical half-spaces. However, unlike in the classical case, the extreme rays of their polar cones provide external representations containing in general superfluous half-spaces. In this paper, we prove that any tropical polyhedral cone in $R^n$ (also known as ``tropical polytope´´ in the literature) admits an essentially unique minimal external representation. The result is obtained by establishing a (partial) anti-exchange property of half-spaces. Moreover, we show that the apices of the half-spaces appearing in such non-redundant external representations are vertices of the cell complex associated with the polyhedral cone. We also establish a necessary condition for a vertex of this cell complex to be the apex of a non-redundant half-space. It is shown that this condition is sufficient for a dense class of polyhedral cones having ``generic extremities´´.