On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908788509900800 |
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| author | Rubin, Natan |
| author_facet | Rubin, Natan |
| contents | 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 ${\mathbb R}^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17913 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions Rubin, Natan Combinatorics Computational Geometry 52A35, 52C35, 05D10 G.2.1; F.2.2 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 ${\mathbb R}^3$. |
| title | On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions |
| topic | Combinatorics Computational Geometry 52A35, 52C35, 05D10 G.2.1; F.2.2 |
| url | https://arxiv.org/abs/2601.17913 |