Here is a variation on the classical polygon illumination problem. For $c \geq 0$ we say that a mirror has reflection index $c$ if whenever a ray hits the mirror with angle of incidence $\alpha$ then the angle of reflection is $$\alpha'=\begin{cases}c\alpha+(1-c)\frac{\pi}{2} & c\le 1 \\\ c^{-1}\alpha & c\geq 1 \end{cases}$$.

The classical problem is : Let $P$ be a polygon and consider the sides as mirrors. Can we place a source of light in the interior of $P$ which illuminates all of $P$? The answer is not known even if we relax "all of $P$" to "$P$ minus a finite set of points".

If we fix $P$ with $n$ sides, the possible reflection indices we can assign to the sides are parametrized by $\mathbb R_{+}^n$. Can we always find an $x\in \mathbb R_{+}^n$ for which the corresponding "perturbed" polygon can be illuminated by some interior light source? Can we choose $x$ close to $(1,1,\dots,1)$?