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Topology of Geometric Joins

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.

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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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Correspondence to Andreas F. Holmsen.

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Editor in charge: János Pach.

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Bárány, I., Holmsen, A.F. & Karasev, R. Topology of Geometric Joins. Discrete Comput Geom 53, 402–413 (2015). https://doi.org/10.1007/s00454-015-9665-2

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Keywords

  • Colorful point sets
  • Colorful Carathéodory theorem
  • Nerve theorem