Billiard obstacles with hidden sets

The Penrose unilluminable room and the Livshits billiard are billiard systems based on a semi-ellipse with a hidden set - a set from which all trajectories are trapped and will never escape. They are counterexamples to the trapped set problem and the illumination problem respectively. In this paper we construct a large class of planar billiard obstacles, not necessarily featuring ellipses, that have the same property. The main result is that for any convex set $\mathcal{H}$ (with some smoothness conditions), we can construct a billiard table $K$ such that trajectories leaving $\mathcal{H}$ always return to $\mathcal{H}$ after one reflection. As corollaries, we give a more general answer to the illumination problem and the trapped set problem.