(Non)existence of mirrors with more than two foci Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes through one of the points $x_i$ it automatically passes through the other two after a number of reflections on $M$?
The question needs some formalisation for what is meant by $M \subset \mathbb{R}^d$ and “rays of light”, although it is easy to guess what they should be:


*

*$M$ is a $(d-1)$-dimensional differentiable submanifold of $\mathbb{R}^d$.

*A ray of light is a continuous function $g:\mathbb{R} \rightarrow \mathbb{R}^d$ for which $g^{-1}(g(\mathbb{R}) \cap M)$ is a discrete set. Moreover, $g$ is locally smooth, linear and unit speed at any point $t\in \mathbb{R}$ for which $g(t) \notin M$: $\left\|g'(t)\right\|=1$ and g''(t)=0. At any point $t^* \in \mathbb{R}$ for which $g(t^*)\in M$ we require $g$ to satisfy the reflection law: denoting $a=\lim_{t \to t^* -} g'(t)$ and $b = \lim_{t \to t^* +} g'(t)$
\begin{equation}
b + a \perp n(g(t))
\end{equation}
and
\begin{equation}
b - a \propto n(g(t))
\end{equation}
(n is a local unit-normal vector field on $M$).
 A: This is not an answer, but suggests one avenue to explore.
There are "$n$-foci ellipses,"
but I am not familiar with their reflection properties.
This image is suggestive (but certainly not definitive):




(Image from Mohd. Javed Khilji, "Multi Foci Closed Curves."
(PDF download link).)


Another reference to these ellipses may be found at this MSE question.
A: Here is a way that works for a set of full measure in any dimension, for any number of points, not just $3$. We can construct it for dimension $2$, then rotate it about the $x$-axis.
Let the points be $x_i=(i,0)$ for $i=1,...,n$. Attach semi-ellipses $\gamma_i$ whose major axes and end points are along the $x$-axis whose foci are $x_i,x_{i+1}$ so that for $i$ odd we take the intersection of the ellipse with the upper half plane, and for $i$ even we take the intersection with the lower half plane. Extend this by adding semicircles $\gamma_0$ and $\gamma_n$ about the end points. If we choose the semiellipses so that the total distance to $x_i$ and $x_{i+1}$ is $2$, and use semicircles with radius $1/2$, this is even a simple closed curve, although there are points where it is not smooth.
Light passing through $x_1$ going up bounces off $\gamma_1$ and is reflected to $x_2$ going down, so it reflects off $\gamma_2$ to $x_3$ going up, etc. After it passes through $x_n$ it bounces off $\gamma_n$ and reverses. Light passing through $x_1$ going down bounces off $\gamma_0$ and reverses through $x_1$, going up. The behavior of the horizontal ray through the nonsmooth points and all $x_i$ is not really defined, but you could argue that it passes through all of the points, too.

Mathematica code for drawing this picture:
ell[i_, x_] := (-1)^(i + 1) Sqrt[3/4 (1 - (x - (i + 1/2))^2)]
circ[i_, x_] := Sqrt[1/4 - (x - i)^2]
Plot[{-circ[1, x], ell[1, x], ell[2, x], ell[3, x], ell[4, x], circ[5, x]}, {x, 0, 6}, 
    PlotRange -> {-1.5, 1.5},  AspectRatio -> 1/2]

