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The illumination problem is a popular topic in recreational mathematics: In amirrored room, is every region illuminable from every point in the region?So-called \enquote{unilluminable rooms} are related to \enquote{trapped sets}in inverse scattering, and to billiards with divided phase space in dynamicalsystems. In each case, a billiard with a semi-ellipse has always been putforward as the standard counterexample: namely the Penrose room, the Livshitsbilliard, and the Bunimovich mushroom respectively. In this paper, we constructa large class of planar billiard obstacles, not necessarily featuring ellipses,that have dark regions, hidden sets, or a divided phase space. The main resultis that for any convex set $\mathcal{H}$, we can construct a convex, everywheredifferentiable billiard table $K$ (at any distance from $\mathcal{H}$) suchthat trajectories leaving $\mathcal{H}$ always return to $\mathcal{H}$ afterone reflection. This billiard generalises the Bunimovich mushroom. Ascorollaries, we give more general answers to the illumination problem and thetrapped set problem. We use recent results from nonsmooth analysis and convexfunction theory, to ensure that the result applies to all convex sets.

Unilluminable rooms, billiards with hidden sets, and Bunimovich mushrooms