Protecting convex sets

A point-set S is protecting a collection F = T₁, T₂,...,_n of n mutually disjoint compact sets if each one of the sets T_i is visible from at least one point in S; thus, for every set T_i ∈F there are points x S and yT_i such that the line segment joining x to y does not intersect any element in F other than T_i. In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F of n mutually disjoint compact convex sets. For an isothetic family F, consisting of n mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. If F is a family of triangles, [4 n/7] points are always sufficient. To protect families of n homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.