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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 AsGet 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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