Tropical convex hulls of infinite sets

In this paper we study the interplay between tropical convexity and its classical counterpart. In particular, we focus on the tropical convex hull of convex sets and polyhedral complexes. %First we investigate the structure of the tropical convex hull of any set and then convex sets in particular. We give a vertex description of the tropical convex hull of a line segment and of a ray in $\mathbb{R}^n/\mathbb {R}\mathbf{1}$ and show that tropical convex hull and classical convex hull commute in $\mathbb{R}^3/\mathbb {R}\mathbf{1}$. Finally we prove results on the dimension of tropically convex fans and give a lower bound on the degree of a tropical curve under certain hypotheses.