Illuminating high-dimensional convex sets

A set L of points in the d-space E^dis said to illuminate a family F={S₁, ..., S_n} of n disjoint compact sets in E^dif for every set S_iin F and every point x in the boundary of S_ithere is a point v in L such that v illuminates x, i.e. the line segment joining v to x intersects the union of the elements of F in exactly {x}. The problem we treat is the size of a set S needed to illuminate a family F={S₁, ..., S_n} of n disjoint compact sets in E^d. We also treat the problem of putting these convex sets in mutually disjoint convex polytopes, each one having at most a certain number of facets.