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Colorful Intersections and Tverberg Partitions

Authors Michael Gene Dobbins , Andreas F. Holmsen, Dohyeon Lee

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Author Details

Michael Gene Dobbins
  • Department of Mathematics and Statistics, Binghamton University, NY, USA
Andreas F. Holmsen
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Dohyeon Lee
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea

Funding

  • Dobbins, Michael Gene: Support from KAIST Advanced Institute for Science-X (KAI-X).
  • Holmsen, Andreas F.: Supported by the Institute for Basic Science (IBS-R029-C1).
  • Lee, Dohyeon: Supported by the Institute for Basic Science (IBS-R029-C1).

Cite As Get BibTex

Michael Gene Dobbins, Andreas F. Holmsen, and Dohyeon Lee. Colorful Intersections and Tverberg Partitions. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 52:1-52:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.52

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 ≥ m ≥ 1 and k a prime power. Suppose F₁, F₂, … , F_m are families of convex sets in ℝ^d, each of size n > (d/m+1)(k-1), such that for any choice C_i ∈ F_i we have ⋂_{i = 1}^m C_i ≠ ∅. 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 ℝ^d that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Topology
Keywords
  • Tverberg’s theorem
  • geometric transversals
  • topological combinatorics
  • configuration space/test map
  • discrete Morse theory

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References

  1. Imre Bárány. A generalization of Carathéodory’s theorem. Discrete Mathematics, 40(2-3):141-152, 1982. Google Scholar
  2. Imre Bárány. Combinatorial convexity, volume 77. American Mathematical Soc., 2021. Google Scholar
  3. Imre Bárány and Gil Kalai. Helly-type problems. Bulletin of the American Mathematical Society, 59(4):471-502, 2022. Google Scholar
  4. Anders Björner. Topological methods. Handbook of combinatorics, 2:1819-1872, 1995. Google Scholar
  5. Jesús De Loera, Xavier Goaoc, Frédéric Meunier, and Nabil Mustafa. The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg. Bulletin of the American Mathematical Society, 56(3):415-511, 2019. Google Scholar
  6. Jürgen Eckhoff. Helly, Radon, and Carathéodory type theorems. In Handbook of convex geometry, pages 389-448. Elsevier, 1993. Google Scholar
  7. Robin Forman. A discrete Morse theory for cell complexes. In Geometry, topology, and physics, Conf. Proc. Lecture Notes Geom. Topology, IV, pages 112-125, 1995. Google Scholar
  8. Andreas Holmsen and Rephael Wenger. Helly-type theorems and geometric transversals. In Handbook of discrete and computational geometry, pages 91-123. Chapman and Hall/CRC, 2017. Google Scholar
  9. Jakob Jonsson. Simplicial complexes of graphs, volume 1928. Springer, 2008. Google Scholar
  10. Dimitry Kozlov. Combinatorial algebraic topology, volume 21. Springer Science & Business Media, 2008. Google Scholar
  11. Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin. Further consequences of the colorful Helly hypothesis. Discrete & Computational Geometry, 63(4):848-866, 2020. Google Scholar
  12. Jiri Matousek. Lectures on discrete geometry, volume 212. Springer Science & Business Media, 2013. Google Scholar
  13. Jiří Matoušek, Anders Björner, Günter M Ziegler, et al. Using the Borsuk-Ulam theorem: lectures on topological methods in combinatorics and geometry, volume 2003. Springer, 2003. Google Scholar
  14. L Montejano. Transversals, topology and colorful geometric results. Geometry—Intuitive, Discrete, and Convex: A Tribute to László Fejes Tóth, pages 205-218, 2013. Google Scholar
  15. Luis Montejano and Roman N Karasev. Topological transversals to a family of convex sets. Discrete & Computational Geometry, 46(2):283-300, 2011. Google Scholar
  16. Sherry Sarkar and Pablo Soberón. Tolerance for colorful Tverberg partitions. European Journal of Combinatorics, 103:103527, 2022. Google Scholar
  17. Helge Tverberg. A generalization of Radon’s theorem. Journal of the London Mathematical Society, 1(1):123-128, 1966. Google Scholar
  18. A Yu Volovikov. On a topological generalization of the Tverberg theorem. Mathematical Notes, 59(3):324-326, 1996. Google Scholar
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