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Mathematician Ernst Straus wondered if a room lined with mirror can always be lit with a single match. He (or someone else) discovered that in the following room, light shone from A can't reach B:

Straus's example

The bad thing here is that light shone at middle segment can approach B arbitrarily close:

Straus's example with lights

So, assuming that reflection at corner is undefined, I think light shone at A can reach all points inside the room except B and only B. (If there were other points that can't be illuminated, the author would have specified, especially if they had an $\epsilon$-ball around them that can't be illuminated.)

This is rather disappointing to me, for the problem itself was born out of a physical motivation. So hence here are my questions:

  1. Is there a smooth ($C^1$) mirror-lined planar room such that a light source fails to illuminate a region in the room in which an $\epsilon$-ball can be fitted inside?

  2. What about the same question as above, with the difference being that we now consider a piecewise $C^1$ curve? Here, we do not define reflection of light at discontinuous points.

  3. What about the same problems worked on higher dimensional smooth surfaces?

  4. What about the same problems worked out on noneuclidean space? It would be interesting if all the questions above failed but it does work on a hyperbolic space.

Necessary rigor can be imbued on the above problems, though it takes some work...

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mathworld.wolfram.com/IlluminationProblem.html: "In 1958, a young Roger Penrose used the properties of the ellipse to describe a room with curved walls that would always have dark (unilluminated) regions, regardless of the position of the candle." –  Rahul Apr 14 '12 at 6:40
oh thanks! I vaguely remembered that I saw something like this before. –  progressiveforest Apr 14 '12 at 6:47
what about $C^\infty$ simple connected curve? hmm.. –  progressiveforest Apr 14 '12 at 7:02

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