Pl\"ucker-type inequalities for mixed areas and intersection numbers of curve arrangements

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