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arXiv · 2403.14909

Colorful Intersections and Tverberg Partitions

Michael Gene Dobbins·Andreas F. Holmsen·Dohyeon Lee
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Abstract

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.

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Michael Gene Dobbins, Andreas F. Holmsen, Dohyeon Lee. 2024-03-22. Colorful Intersections and Tverberg Partitions. https://arxiv.org/abs/2403.14909

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