Separating convex sets by straight lines (Q5951952)

A family of pairwise disjoint sets in the plane is said to be separable, if any two sets can be separated by a straight line which does not intersect any member of the family. According to the first theorem of this paper there is a family of infinitely many pairwise disjoint circular disks (or squares) in the plane which has no separable subfamily consisting of three sets. This is related to a construction of K. Villanger concerning straight line segments which is described in a paper of \textit{H. Tverberg} [Math. Scand. 45, 255-260 (1979; Zbl 0431.52003)]. It is also shown that there is no such family of sets with bounded inradius and circumradius. NEWLINENEWLINENEWLINEThe first theorem contrasts with the second one which says that every uncountable family of pairwise disjoint convex sets in the plane has two uncountable subfamilies that can be separated by a straight line. The original proof of this was greatly simplified by V. Totik. Finally it is shown that there exists an infinite family of pairwise disjoint unit segments in 3-space such that there are no two members that can be separated from a third by a plane.