Colorful Intersections and Tverberg Partitions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916172046270464 |
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| author | Dobbins, Michael Gene Holmsen, Andreas F. Lee, Dohyeon |
| author_facet | Dobbins, Michael Gene Holmsen, Andreas F. Lee, Dohyeon |
| contents | The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime power. Suppose $F_1, F_2, \dots, F_m$ are families of convex sets in $\mathbb{R}^d$, each of size $n > (\frac{d}{m}+1)(k-1)$, such that for any choice $C_i\in F_i$ we have $\bigcap_{i=1}^mC_i\neq \emptyset$. Then, one of the families $F_i$ admits a Tverberg $k$-partition. That is, one of the $F_i$ can be partitioned into $k$ nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning $r$-dimensional transversals to families of convex sets in $\mathbb{R}^d$ that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Colorful Intersections and Tverberg Partitions Dobbins, Michael Gene Holmsen, Andreas F. Lee, Dohyeon Combinatorics Metric Geometry The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime power. Suppose $F_1, F_2, \dots, F_m$ are families of convex sets in $\mathbb{R}^d$, each of size $n > (\frac{d}{m}+1)(k-1)$, such that for any choice $C_i\in F_i$ we have $\bigcap_{i=1}^mC_i\neq \emptyset$. Then, one of the families $F_i$ admits a Tverberg $k$-partition. That is, one of the $F_i$ can be partitioned into $k$ nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning $r$-dimensional transversals to families of convex sets in $\mathbb{R}^d$ that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano. |
| title | Colorful Intersections and Tverberg Partitions |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2403.14909 |