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Any collection of $n$ compact convex planar sets $K_1,\dots, K_n$ defines avector of ${n\choose 2}$ mixed areas $V(K_i,K_j)$ for $1\leq i<j\leq n$. Weshow that for $n\geq 4$ these numbers satisfy certain Pl\"ucker-typeinequalities. Moreover, we prove that for $n=4$ these inequalities completelydescribe 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 offull dimension. As an application, we obtain an inequality description for thesmallest positive homogeneous set containing the configuration space ofintersection numbers of quadruples of tropical curves.

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