# Polygons and mirrors

Given a point $A$ inside a non-convex polygon $P$, is it always possible to place a finite set of mirrors (not necessarily along the boundary of $P$, any position inside $P$ is allowed) given by straight segments such that every point of $P$ is visible from $A$? (Light-rays propagate along straight lines, are reflected in the usual way on both sides of mirrors and are absorbed by the boundary of $P$.)

Is there a position of mirrors which is "universal" in the sense that any pair of points inside $P$ are within view of each other?

These questions have of course obvious higher-dimensional generalizations.

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The boundary of the polygon can of course always be supposed to be covered by mirrors (and it reflects then light-rays in the same ways as a polygonal billiard) without loss of generality. –  Roland Bacher Oct 13 '10 at 9:26

These are rooms all of whose walls are mirrors but which have a pair of points $x$ and $y$ such that if a light is placed at $x$, point $y$ is dark. It is an unresolved conjecture that, at least in rational polygons (angles rational multiples of $\pi$), the number of dark points for any light position is finite. If that conjecture were true, then it seems plausible that it would be possible to place additional internal mirrors to cover those points and still illuminate the remainder of the room.
I am not completely convinced that one can solve the problem with mirrors for a finite number of dark points. Indeed, mirrors in the interior of the polygon can create new invisible points and regions. Putting a mirror in the interior of a polygon $P$ corresponds to branching in the developed surface with two ramifications of order two at the endpoints of the mirrors. –  Roland Bacher Oct 14 '10 at 7:32