  We study the relationship between min-plus, max-plus and Euclidean convexity
for subsets of $\mathbb{R}^n$. We introduce a construction which associates to
any max-plus convex set with compact projectivisation a canonical matrix called
its dominator. The dominator is a Kleene star whose max-plus column space is
the min-plus convex hull of the original set. We apply this to show that a set
which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii)
a Euclidean polytope must also be the third. In particular, these results
answer a question of Sergeev, Schneider and Butkovic and show that row spaces
of tropical Kleene star matrices are exactly the "polytropes" studied by Joswig
and Kulas.
