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arXiv · 2411.07241

A necessary and sufficient condition for $k$-transversals

Daniel McGinnis·Nikola Sadovek
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Abstract

We solve a long-standing open problem posed by Goodman \& Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal for any $0 \le k \le d-1$. This result is a common generalization of Helly's theorem ($k=0$) and the Goodman-Pollack-Wenger theorem ($k=d-1$). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex $k$-transversal to a family of convex sets in $\mathbb{C}^d$, extending the work of McGinnis ($k=d-1$). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of \v{Z}ivaljevi\'c-Vre\'cica and Dol'nikov in the real case and of Sadovek-Sober\'on in the complex case.

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Daniel McGinnis, Nikola Sadovek. 2024-11-11. A necessary and sufficient condition for $k$-transversals. https://arxiv.org/abs/2411.07241

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