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There is a long-unsolved problem posed by Janos Pach, sometimes known as the enchanted forest problem, which asks if it is possible to block a point light source in the plane from reaching infinity by disjoint mirrored disks.1


          MirrorTrees
          Image from the MO question, "Trapped rays bouncing between two convex bodies."
            (See also "Lightray trapped between two mirror disks: Computation formulation?.")

A decade later, a student (Octavia Petrovici) and I posed a seemingly easier problem, asking whether it is possible to block light with mirror segments;2 but it remains open (and Octavia and I conjectured the answer is No).

It occurred to me it might be easier to answer this question (positively):

Q. Is it possible to block light from a single point light source in the plane by a finite collection of mirrored disjoint convex bodies?

One version insists that the convex bodies be smooth, so that lightray reflection is well-defined from every point. However, one can define reasonable reflection rules from, e.g., convex polygons. Because no version is settled, there is some "problem-designer" freedom here.

The only novelty in Q is that one might imagine "engineering" the convex reflectors individually to achieve the desired goal, rather than being constrained to only use disks, or only use segments. Still, all these reflectors are dispersive, as I learned from another MO question, "Are rounded rectangle billiard dynamics ergodic?," and it is difficult for me to imagine how one could block all paths with dispersive obstacles...


*1 H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer, New York, 1991. p.19. (Springer link)

*2 "Trapping Light Rays with Segment Mirrors." 2001.

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  • $\begingroup$ Is there a variant of this in the two-dimensional space? $\endgroup$ – Zsbán Ambrus Feb 2 '14 at 9:23
  • $\begingroup$ @ZsbánAmbrus: Oh, sorry, I meant all this to be in $\mathbb{R}^2$, i.e., in the Euclidean plane. Now edited to be clearer. $\endgroup$ – Joseph O'Rourke Feb 2 '14 at 14:37
  • $\begingroup$ It is unlike even for convex bodies. It should be possible to prove it in dimension $(2+\epsilon).$ This is because this problem seems somehow related with the transience of random walks. That it is false in dim $2$ but true for higher dimensions. This is just a felling of the problem after some thought. $\endgroup$ – user39115 May 3 '16 at 19:19
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    $\begingroup$ This reminds me of light trapping methods by random textures used in solar cells: aip.scitation.org/doi/abs/10.1063/1.339189 $\endgroup$ – Tobias Fritz Sep 20 '17 at 13:22
  • $\begingroup$ Has anyone tried to approach this via contact geometry? Perhaps there is some theorem or conjecture in contact geometry that would resolve this? $\endgroup$ – Tobias Fritz Sep 20 '17 at 13:23

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