  Tropical algebraic geometry is the geometry of the tropical semiring (R, min,
+). The theory of total positivity is a natural generalization of the study of
matrices with all minors positive. In this paper we introduce the totally
positive part of the tropicalization of an arbitrary affine variety, an object
which has the structure of a polyhedral fan. We then investigate the case of
the Grassmannian, denoting the resulting fan \Trop^+ Gr_{k,n}. We show that
\Trop^+ Gr_{2,n} is the Stanley-Pitman fan, which is combinatorially the fan
dual to the (type A_{n-3}) associahedron, and that \Trop^+ Gr_{3,6} and \Trop^+
Gr_{3,7} are closely related to the fans dual to the types D_4 and E_6
associahedra. These results are reminiscent of the results of Fomin and
Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster
algebra structure which is of types A_{n-3}, D_4, and E_6 for Gr_{2,n},
Gr_{3,6}, and Gr_{3,7}. We suggest a general conjecture about the positive part
of the tropicalization of a cluster algebra.
