  Motivated by the construction of the free Banach lattice generated by a
Banach space, we introduce and study several vector and Banach lattices of
positively homogeneous functions defined on the dual of a Banach space $E$. The
relations between these lattices allow us to give multiple characterizations of
when the underlying Banach space $E$ is finite-dimensional and when it is
reflexive. Furthermore, we show that lattice homomorphisms between free Banach
lattices are always composition operators, and study how these operators behave
on the scale of lattices of positively homogeneous functions.
