  Let $P_1, P_2,\ldots, P_{d+1}$ be pairwise disjoint $n$-element point sets in
general position in $d$-space. It is shown that there exist a point $O$ and
suitable subsets $Q_i\subseteq P_i \; (i=1, 2, \ldots, d+1)$ such that
$|Q_i|\geq c_d|P_i|$, and every $d$-dimensional simplex with exactly one vertex
in each $Q_i$ contains $O$ in its interior. Here $c_d$ is a positive constant
depending only on $d$.
