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

TL;DR

This paper establishes Pl"ucker-type inequalities for mixed areas of convex planar sets, characterizes their solution space for four sets, and explores their implications for tropical curve intersection numbers.

Contribution

It introduces new inequalities for mixed areas, fully describes the solution space for four sets, and links these results to tropical geometry intersection numbers.

Findings

Mixed areas satisfy Pl"ucker-type inequalities for n≥4.

For n=4, these inequalities fully characterize the mixed area vectors.

The solution space has a semialgebraic closure of full dimension for n≥4.

Abstract

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.