Quantitative Transversal Theorems in the Plane

Abstract

Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent $k$-ordering, for the existence of a hyperplane transversal for sets in ${\mathbb{R}}^d$. We obtain a quantitative generalization of Hadwiger's theorem in ${\mathbb{R}}^2$, showing that compact convex sets in ${\mathbb{R}}^2$ with a quantitative version of consistent ordering have a transversal satisfying quantitative requirements. Our proof generalizes the methods in Wenger's proof of Hadwiger's theorem in ${\mathbb{R}}^2$. We also prove colorful versions of our results.

23 pages, 11 figures