Quantitative Transversal Theorems in the Plane

Abstract: Hadwiger's theorem is a variant of Helly-type theorems involving common transversals to families of convex sets instead of common intersections. In this paper, we obtain a quantitative version of Hadwiger's theorem on the plane: given an ordered family of pairwise disjoint and compact convex sets in $\mathbb{R}^2$ and any real-valued monotone function on convex subsets of $\mathbb{R}^2,$ if every three sets have a common transversal, respecting the order, such that the intersection of the sets with each half-plane defined by the transversal are valued at least (or at most) some constant $\alpha,$ then the entire family has a common transversal with the same property. Unlike previous generalizations of Hadwiger's theorem, we prove that disjointness is necessary for the quantitative case. We also prove colorful versions of our results.