Quantitative Transversal Theorems in the Plane

TL;DR

This paper extends Hadwiger's theorem to a quantitative setting in the plane, providing conditions under which convex sets have transversals meeting specific quantitative criteria, and introduces colorful variants of these results.

Contribution

It offers a quantitative generalization of Hadwiger's theorem in , incorporating consistent orderings and colorful versions, advancing the understanding of transversals for convex sets.

Findings

Established a quantitative version of Hadwiger's theorem in .

Proved that sets with a quantitative consistent ordering have suitable transversals.

Developed colorful variants of the main theorems.

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.