A complex analogue of the Goodman-Pollack-Wenger theorem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929213431349248 |
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| author | McGinnis, Daniel |
| author_facet | McGinnis, Daniel |
| contents | A \textit{$k$-transversal} to family of sets in $\mathbb{R}^d$ is a $k$-dimensional affine subspace that intersects each set of the family. In 1957 Hadwiger provided a necessary and sufficient condition for a family of pairwise disjoint, planar convex sets to have a $1$-transversal. After a series of three papers among the authors Goodman, Pollack, and Wenger from 1988 to 1990, Hadwiger's Theorem was extended to necessary and sufficient conditions for $(d-1)$-transversals to finite families of convex sets in $\mathbb{R}^d$ with no disjointness condition on the family of sets. We prove an analogue of the Goodman-Pollack-Wenger theorem in the complex setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_16467 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A complex analogue of the Goodman-Pollack-Wenger theorem McGinnis, Daniel Combinatorics Metric Geometry 52A35 A \textit{$k$-transversal} to family of sets in $\mathbb{R}^d$ is a $k$-dimensional affine subspace that intersects each set of the family. In 1957 Hadwiger provided a necessary and sufficient condition for a family of pairwise disjoint, planar convex sets to have a $1$-transversal. After a series of three papers among the authors Goodman, Pollack, and Wenger from 1988 to 1990, Hadwiger's Theorem was extended to necessary and sufficient conditions for $(d-1)$-transversals to finite families of convex sets in $\mathbb{R}^d$ with no disjointness condition on the family of sets. We prove an analogue of the Goodman-Pollack-Wenger theorem in the complex setting. |
| title | A complex analogue of the Goodman-Pollack-Wenger theorem |
| topic | Combinatorics Metric Geometry 52A35 |
| url | https://arxiv.org/abs/2303.16467 |