Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unilluminable rooms, billiards with hidden sets, and Bunimovich
  mushrooms   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.

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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th