A necessary and sufficient condition for $k$-transversals
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| Format: | Preprint |
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2024
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| _version_ | 1866909998258323456 |
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| author | McGinnis, Daniel Sadovek, Nikola |
| author_facet | McGinnis, Daniel Sadovek, Nikola |
| contents | 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 Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A necessary and sufficient condition for $k$-transversals McGinnis, Daniel Sadovek, Nikola Combinatorics Metric Geometry 52A35 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 Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case. |
| title | A necessary and sufficient condition for $k$-transversals |
| topic | Combinatorics Metric Geometry 52A35 |
| url | https://arxiv.org/abs/2411.07241 |