  When $G$ is a complex reductive algebraic group, MV polytopes are in
bijection with the non-negative tropical points of the unipotent group of $G$.
By fixing $w$ from the Weyl group, we can define MV polytopes whose highest
vertex is labelled by $w$. We show that these polytopes are in bijection with
the non-negative tropical points of the reduced double Bruhat cell labelled by
$w^{-1}$. To do this, we define a collection of generalized minor functions
$\Delta_\gamma^\text{new}$ which tropicalize on the reduced Bruhat cell to the
BZ data of an MV polytope of highest vertex $w$.
  We also describe the combinatorial structure of MV polytopes of highest
vertex $w$. We explicitly describe the map from the Weyl group to the subset of
elements bounded by $w$ in the Bruhat order which sends $u \mapsto v$ if the
vertex labelled by $u$ coincides with the vertex labelled by $v$ for every MV
polytope of highest vertex $w$. As a consequence of this map, we prove that
these polytopes have vertices labelled by Weyl group elements less than $w$ in
the Bruhat order.
