  In this paper we study general tropical linear spaces locally: For any basis
B of the matroid underlying a tropical linear space L, we define the local
tropical linear space L_B to be the subcomplex of L consisting of all vectors v
that make B a basis of maximal v-weight. The tropical linear space L can then
be expressed as the union of all its local tropical linear spaces, which we
prove are homeomorphic to Euclidean space. Local tropical linear spaces have a
simple description in terms of polyhedral matroid subdivisions, and we prove
that they are dual to mixed subdivisions of Minkowski sums of simplices. Using
this duality we produce tight upper bounds for their f-vectors. We also study a
certain class of tropical linear spaces that we call conical tropical linear
spaces, and we give a simple proof that they satisfy Speyer's f-vector
conjecture.
