Quantitative Transversal Theorems in the Plane

TL;DR

The paper develops a quantitative, planar generalization of Hadwiger's transversal theorem by introducing (f,α)-transversals for convex sets, where f measures a notion of largeness and α sets a threshold. It defines (f,α)-consistent orderings and (f,α)-stabbing directions to extend Wenger's Hadwiger proof strategy to a quantitative setting, and proves a planar quantitative Hadwiger theorem under the ordering assumption. It further establishes stabbing-order variants and a colorful extension, showing that under appropriate conditions certain colors admit a common transversal. The work outlines higher-dimensional conjectures, aiming to generalize these notions to (f,α)-hyperplane transversals and consistent k-orderings in R^d, with potential broad impact on quantitative Helly-type and transversal theory.

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.

Figures (11)

• Figure 1: Left: an $(f,\alpha)$-transversal for three convex sets. Right: an $(f,\alpha)$-separation line for two convex sets with the ellipse on the left side and the quadrilateral on the right side. (Remember that $\alpha$ represents a vector assigning an $\alpha_i$ value to each set, with these values potentially differing by set.)
• Figure 2: Convex sets $C_1,...,C_4$ with subsets $C_1',...,C_4'$ shaded in. Projections onto the perpendicular line are drawn, and the $(f,\alpha)$-transversal is going through the intersection point.
• Figure 3: Sets $A$ and $B$, along with a point $p\in \ell$ and a line $\ell_\theta$, obtained by rotating $\ell$ clockwise. The darker shaded area of $A$ is $A'=\ell^-(A)$ and the darker shaded area of $B$ is $B'=\ell^+(B)$.
• Figure 4: Sets with nonempty intersection of $S_1$ and $S_2$. The line $\ell$ separates $A$ and $B$, and the line $\ell'$ separates $X$ and $Y$. Note that no $(f,\alpha)$-transversal exists in the order provided for the sets $X$, $A$, $B$ or $A$, $X$, $Y$.
• Figure 5: The sets $S_1$ (thick blue arcs on the left side of $v_{\text{stab}}$) and $S_2$ (thick red arcs on the right side of $v_{\text{stab}}$) for some family of convex sets drawn on a circle. The arcs of $S_1$ and $S_2$ do not intersect, so we are guaranteed stabbing directions for all sets. We picked one of these to be $v_{\text{stab}}$.

Theorems & Definitions (55)

• Definition 1: Amenta:2017ed
• Theorem 1.1
• Theorem 1.2
• Definition 2
• Remark 2.1
• Definition 3
• Definition 4
• Lemma 2.2
• proof
• Corollary 2.3