Separating convex sets in the plane

Given a set A in R² and a collection S of plane sets, we say that a line L separates A from S if A is contained in one of the closed half-planes defined by L, while every set in S is contained in the complementary closed half-plane. We prove that, for any collection F of n disjoint disks in R², there is a line L that separates a disk in F from a subcollection of F with at least {bottom right crop}(n-7)/4{bottom left crop} disks. We produce configurations H_n and G_n, with n and 2 n disks, respectively, such that no pair of disks in H_n can be simultaneously separated from any set with more than one disk of H_n, and no disk in G_n can be separated from any subset of G_n with more than n disks. We also present a set J_m with 3 m line segments in R², such that no segment in J_m can be separated from a subset of J_m with more than m+1 elements. This disproves a conjecture by N. Alon et al. Finally we show that if F is a set of n disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a line L that separates one of the segments from at least {bottom right crop}n/3{bottom left crop}+1 elements of F.