Piercing intersecting convex sets

TL;DR

This work addresses a Helly-type transversal problem for two finite convex families $\mathcal{A}$ and $\mathcal{B}$ in $\mathbb{R}^3$ with $A\cap B\neq\emptyset$ for all pairs; it proves a line transversal result in the special case where $\mathcal{A}$ and $\mathcal{B}$ consist of vertical polygons lying in parallel planes, guaranteeing a line that intersects all sets of one family. It then strengthens the result by showing a location-restricted fractional transversal: under the same setup, there is a line lying in the plane of some $A_i$ hitting $\frac{1}{6}|\mathcal{B}|$ sets in $\mathcal{B}$ (or symmetrically in a $B_j$-plane hitting $\frac{1}{6}|\mathcal{A}|$). The main technique combines affine normalization, a bilinear feasibility framework via Farkas' lemma, dualizing to a non-constructive existence proof, and a secondary fractional Helly argument to obtain a plane-restricted transversal; a higher-dimensional partial extension is outlined. These results advance understanding of Helly-type line transversals for intersecting convex families and provide a framework for constructive computation in fixed dimension.

Abstract

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.

Figures (3)

• Figure 1: An illustration of the $n=m=3$ case. The red sets form the family $\mathcal{A}$ and the blue sets form the family $\mathcal{B}$.
• Figure 2: The quadrangle $Q$, the lines $\ell_A$ and $\ell_B$, and their intersection at $R$.
• Figure :

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• Remark
• Lemma 3
• proof
• Lemma 4: Fractional Helly for vertical line segments
• proof : Proof of Theorem \ref{['th:fracH']}
• Proposition 5
• proof