  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.
%$\mathbb{R}^3/\mathbb {R}\mathbf{1}$.
  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.
