I am struggling with the following problem.
Given $d+1$ pairs of points $P_i=(x_i, y_i)\in \mathbb{R}^d\times \mathbb{R}^d$, $i=0,\ldots ,d$. Show that we can split the pairs $P_i$ into $u_i$ and $v_i$ such that their convex hulls $\rm{conv}(u_0,\ldots,u_d)\subset \mathbb{R}^d$ and $\rm{conv}(v_0,\ldots,v_d)\subset \mathbb{R}^d$ intersect.
I first thought that it could be somehow shown by using Tverberg or the Carathéodory's colorful Theorem, but I was not able to. Then I thought about proving it via contradiction. That is, assume that for any split the convex hulls do not intersect. Then pick a split for which the distance between the two convex hulls is minimal (it then is >0) and construct another split, which has a smaller distance and thus we obtain a contradiction. Unfortunately, that I couldn't work out either.