  We consider the geometric join of a family of subsets of the Euclidean space.
This is a construction frequently used in the (colorful) Carath\'eodory 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.
