On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions

TL;DR

This work shows that a finite family of $n$ pairwise intersecting convex sets in $R^3$ admits a line transversal hitting $\Theta(n)$ of the sets, resolving a key variant of a problem posed by Martínez, Roldán-Pensado and Rubin. The authors combine a Ramsey-type result for strictly 2-intersecting planar families with a careful analysis of 3D line arrangements and a fractional hyperplane argument to produce a separating plane and a large planar transversal, ultimately yielding a linear-size line transversal. The results offer a rare, nontrivial sufficient condition for large line transversals in three dimensions and connect planar Ramsey phenomena to 3D transversal theory, with implications for related conjectures and higher-dimensional questions.

Abstract

The 1913 Helly's theorem states that any family ${\cal K}$ of $n\geq d+1$ convex sets in ${\mathbb R}^d$ can be pierced by a single point if and only if any $d+1$ of ${\cal K}$'s elements can. In 2002 Alon, Kalai, Matoušek and Meshulam ruled out the possibility of similar criteria for the existence of lines crossing multiple convex sets in dimension $d\geq 3$ -- for any $k\geq 3$, they described arbitrary large families ${\cal K}$ of convex sets in ${\mathbb R}^3$ so that any $k$ elements of ${\cal K}$ can be crossed by a line yet no $k+4$ of them can. Let ${\cal K}$ be a family of $n$ pairwise intersecting convex sets in ${\mathbb R}^3$. We show that there exists a line crossing $Θ(n)$ elements of ${\cal K}$. This resolves the most extensively studied variant of a problem by Martínez, Roldán-Pensado and Rubin (Discrete Comput. Geom. 2020) which was highlighted by Bárány and Kalai (Bull. Amer. Math. Soc. 2021). Our result adds to the very few sufficient (and non-trivial) conditions that have been known for the existence of line transversals to large families of convex sets. Our argument is based on a Ramsey-type result of independent interest for families of pairwise intersecting convex sets in ${\mathbb R}^2$, and the structure of line arrangements in ${\mathbb R}^3$.

Figures (13)

• Figure 1: Left: A projected strictly 2-intersecting family ${{\mathcal{K}}}^*=\{K_1^*,\ldots,K_4^*\}$ in ${\mathbb R}^2$, and its realization by $4$ lines $\ell_1^*,\ldots,\ell^*_4$. Center: The lines $\ell_i$ and $\ell_j$ are separated by the plane $\pi$ -- the vertical line $\ell_{i,j}$ through $\ell_i$ and $\ell_j$ crosses $\pi$ in-between $p(\ell_i,\ell_j)$ and $p(\ell_j,\ell_i)$. Since $p(\ell_i,\ell_j)\in K_i$ and $p(\ell_j,\ell_i)\in K_j$, the intersection $\ell_{i,j}\cap \pi$ must lie in $K_i\cup K_j$. Right: The depicted strictly 2-intersecting family $\{A,B,C,D\}$ in ${\mathbb R}^2$ cannot be realized, in the strong sense detailed above, by any 4 lines.
• Figure 2: Left: The sequence $p_1,\ldots,p_6$ of $n=6$ points forms a 6-cap. Right: A 6-cup of points.
• Figure 3: An $n$-cap and an $n$-cup of lines (resp., left and right).
• Figure 4: The depicted sequence $K_1,\ldots,K_4$ is realized by 4 lines $\ell_1,\ldots,\ell_4$.
• Figure 5: Left: The lines $\ell_1$ and $\ell_2$ satisfy $\ell_1\succ \ell_2$. Depicted are the parallel planes $\pi(\ell_1,\ell_2)\supset \ell_1$ and $\pi(\ell_2,\ell_1)\supset \ell_2$, the unique vertical line $\ell_{1,2}$ through $\ell_1$ and $\ell_2$, and the points $p(\ell_1,\ell_2)=\ell_{1,2}\cap \ell_1$ and $p(\ell_2,\ell_1)\in \ell_{1,2}\cap \pi(\ell_2,\ell_1)$. Right: The sequence of 4 lines in ${\mathbb R}^3$ is monotone, for we have that $\ell_1\succ \ell_2\succ\ell_3\succ \ell_4$, and the projections $\ell_1^*,\ldots,\ell^*_4$ form a 4-cap.

Theorems & Definitions (33)

• Theorem 1.1: Fractional Helly's Theorem KatchalskiLiu
• Theorem 1.2: The $(p,q)$-theorem AlonKleitman
• Theorem 1.3: Lovász's Colorful Helly's Theorem 1982 ImreColored
• Theorem 1.4: The $(p,q)$-theorem for hyperplanes AlonKalai
• Theorem 1.5: A fractional Helly's theorem for hyperplanes AlonKalai
• Conjecture 1.6
• Conjecture 1.7
• Theorem 1.8
• Theorem 2.1: ErdSzRamsey
• Theorem 2.2