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 Plücker-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.