Plücker-type inequalities for mixed areas and intersection numbers of curve arrangements

Mixed areas and mixed volumes are central objects in the measure-theoretic convexity theory. They also play a fundamental role for applications in stochastic, algebraic and tropical geometry, with the link to algebraic and tropical geometry established via the celebrated BKK theorem. Still, some very basic questions on the relations between mixed volumes within a collection of convex bodies remain unsolved. For example, already the exact relations of the 10 mixed areas $V(K_i,K_j)$, $1 \le i \le j \le 4$, of four planar convex bodies $K_1,...,K_4$ are not known and it is not even know if the exact relations are semi-algebraic.

Any collection of $n$ compact convex planar sets $K_1,\dots, K_n$ defines a vector of ${n\choose 2}$ pure mixed areas $V(K_i,K_j)$ for $1\leq i<j\leq n$, where ``pure&#039;&#039; means that i is not equal to j. 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 pure 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. <br />
This is a joint work with Ivan Soprunov (see arxiv.org/abs/2112.13128).