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.
From: Yuki Takahashi [view email]
[v1]
Mon, 21 Aug 2023 20:25:06 UTC (701 KB)
[v2]
Mon, 1 Dec 2025 23:21:11 UTC (593 KB)
[v3]
Fri, 3 Jul 2026 21:05:47 UTC (593 KB)
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