Tropical convex hulls of polyhedral sets

Abstract

In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropically convex polyhedra in any dimension. %${\mathbb{R}}^3/\mathbb{R}1$. Finally we show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.

arXiv comment12 pages, 3 Figures, New version with corrected proof of Theorem 1.10 and further results on tropically convex polyhedra

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• The Minkowski Theorem for Max-plus Convex Sets
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• Weighted digraphs and tropical cones
• The tropical double description method
• Tropical Geometry and Mechanism Design
• Face posets of tropical polyhedra and monomial ideals