Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that 2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, n/2] points are always sufficient and n/2] points are sometimes necessary to protect it. IfF is a family of triangles, 4n/7] points are always sufficient. To protect families ofn homothetic triangles, n/2] points are always sufficient and n/2] points are sometimes necessary.