A linear worst-case lower bound on the number of holes inside regions visible due to multiple diffuse reflections

It is known that the region V(s) of a simple polygon P, directly visible (illuminable) from an internal point s, is simply connected. Aronov et al. 2] established that the region V₁(s) of a simple polygon visible from an internal point s due to at most one diffuse reflection on the boundary of the polygon P, is also simply connected. In this paper we establish that the region V₂(s), visible from s due to at most two diffuse reflections may be multiply connected; we demonstrate the construction of an n-sided simple polygon with a point s inside it so that the region of P visible from s after at most two diffuse reflections is multiply connected. We also show that V₃(s), the region of P visible from s after at most three diffuse reflections, can have (n) holes.A part of this work was done when this author was visiting the University of Miami, Coral Gables, Florida, USA.