Separating convex sets by straight lines

We answer some questions of Tverberg about separability properties of families of convex sets. In particular, we show that there is a family of infinitely many pairwise disjoint closed disks, no two of which can be separated from two others by a straight line. No such construction exists with equal disks. We also prove that every uncountable family of pairwise disjoint convex sets in the plane has two uncountable subfamilies that can be separated by a straight line.