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Discrete & Computational Geometry

, Volume 53, Issue 2, pp 402–413 | Cite as

Topology of Geometric Joins

  • Imre Bárány
  • Andreas F. Holmsen
  • Roman Karasev
Article

Abstract

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carathéodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least \(d+1\) sets, where \(d\) is the dimension of the space, then the geometric join is contractible. We are able to prove this when \(d\) equals \(2\) and \(3\), while for larger \(d\) we show that the geometric join is contractible provided the number of sets is quadratic in \(d\). We also consider a matroid generalization of geometric joins and provide similar bounds in this case.

Keywords

Colorful point sets Colorful Carathéodory theorem  Nerve theorem 

Notes

Acknowledgments

The authors are grateful to two anonymous referees for helpful comments and suggestions. I. B. was partially supported by ERC Advanced Research Grant No. 267165 (DISCONV), and by Hungarian National Research Grant K 83767. A. F. H. was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021048). R. K. was supported by the Dynasty foundation.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Imre Bárány
    • 1
    • 2
  • Andreas F. Holmsen
    • 3
  • Roman Karasev
    • 4
    • 5
  1. 1.Alfréd Rényi Mathematical InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Department of MathematicsUniversity College LondonLondonUK
  3. 3.Department of Mathematical SciencesKAISTDaejeonSouth Korea
  4. 4.Department of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  5. 5.Institute for Information Transmission Problems RASMoscowRussia

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