  In classical geometry, a linear space is a space that is closed under linear
combinations. In tropical geometry, it has long been a consensus that tropical
varieties defined by valuated matroids are the tropical analogue of linear
spaces. It is not difficult to see that each such space is tropically convex,
i.e. closed under tropical linear combinations. However, we will also show that
the converse is true: Each tropical variety that is also tropically convex is
supported on the complex of a valuated matroid. We also prove a tropical
local-to-global principle: Any closed, connected, locally tropically convex set
is tropically convex.
