  Any collection of $n$ compact convex planar sets $K_1,\dots, K_n$ defines a
vector of ${n\choose 2}$ mixed areas $V(K_i,K_j)$ for $1\leq i<j\leq n$. We
show that for $n\geq 4$ these numbers satisfy certain Pl\"ucker-type
inequalities. Moreover, we prove that for $n=4$ these inequalities completely
describe the space of all mixed area vectors $(V(K_i,K_j) : 1\leq i<j\leq 4)$.
For arbitrary $n\geq 4$ we show that this space has a semialgebraic closure of
full dimension. As an application, we obtain an inequality description for the
smallest positive homogeneous set containing the configuration space of
intersection numbers of quadruples of tropical curves.
