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arXiv · 1309.0920

Topology of geometric joins

Imre Barany·Andreas F. Holmsen·Roman Karasev
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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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Imre Barany, Andreas F. Holmsen, Roman Karasev. 2015-01-07. Topology of geometric joins. https://doi.org/10.1007/s00454-015-9665-2

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