PROTECTING CONVEX-SETS

A point-set S is protecting a collection F = {T1, T2,..., T(n)} of 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) is-an-element-of F there are points x is-an-element-of S and y is-an-element-of T(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, [4n/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.