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) Carath,odory and Tverberg theorems, and their relatives. We conjecture that when the family has at least sets, where is the dimension of the space, then the geometric join is contractible. We are able to prove this when equals and , while for larger we show that the geometric join is contractible provided the number of sets is quadratic in . We also consider a matroid generalization of geometric joins and provide similar bounds in this case.