Primitive illumination systems for families of convex bodies in the plane (Q1360951)

Let \(\mathcal{F}\) be a family of \(n\) disjoint convex bodies in the plane. A set \(V\) of exterior light sources illuminate \(\mathcal{F}\) if for every boundary point \(u\) of any member of \(\mathcal{F}\) there is a point \(v\) in \(V\) such that the open line segment joining \(u\) and \(v\) is disjoint from the union of the elements of \(\mathcal{F}\). An illumination system \(V\) is called primitive if no proper subset of \(V\) illuminates \(\mathcal{F}\). The authors prove that \(p_{\max }(\mathcal{F)},\) i.e. the maximum number of points in a primitive illumination system, is finite only if all the bodies in \(\mathcal{F}\) are polygons. To be sufficient this condition needs an additional condition about the mutual position of the member of \(\mathcal{F}\). However they show that, in general, this condition is not more necessary. They also study more in detail the case where \((\mathcal{F)}\) consists of three convex polygons. As for the minimum of \(p_{\max }(\mathcal{F)}\) over all the families of \(n\) convex bodies, denoted by \(p_{\max }(n )\), the authors prove that \(p_{\max }(2)=4,\) \( p_{\max }(3)\leq 6\) and \(p_{\max }(n)\leq 3n-2\) for \(n\geq 4\).