# Unilluminable rooms, billiards with hidden sets, and Bunimovich mushrooms

## TL;DR

This paper constructs a broad class of billiard tables with dark regions or divided phase space, generalizing known examples, and provides new insights into the illumination and trapped set problems using convex analysis.

## Contribution

It introduces a method to construct convex billiard tables with hidden sets for any convex obstacle, extending the Bunimovich mushroom model and broadening understanding of illumination and trapped sets.

## Key findings

- Constructed billiard tables with dark regions for any convex obstacle.
- Generalized the Bunimovich mushroom to a wider class of billiards.
- Provided new solutions to the illumination and trapped set problems.

## Abstract

The illumination problem is a popular topic in recreational mathematics: In a mirrored 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 dynamical systems. In each case, a billiard with a semi-ellipse has always been put forward as the standard counterexample: namely the Penrose room, the Livshits billiard, and the Bunimovich mushroom respectively. In this paper, we construct a large class of planar billiard obstacles, not necessarily featuring ellipses, that have dark regions, hidden sets, or a divided phase space. The main result is that for any convex set $\mathcal{H}$, we can construct a convex, everywhere differentiable billiard table $K$ (at any distance from $\mathcal{H}$) such that trajectories leaving $\mathcal{H}$ always return to $\mathcal{H}$ after one reflection. This billiard generalises the Bunimovich mushroom. As corollaries, we give more general answers to the illumination problem and the trapped set problem. We use recent results from nonsmooth analysis and convex function theory, to ensure that the result applies to all convex sets.