# Convex geometry: Under what conditions do ellipsoids have a focal property?

Given an ellipsoid $$E$$, we consider the trajectory that light would make inside if the surface of $$E$$ it would be a mirror In other words, we consider trajectories. $$gamma:[0infty)erect_E[0infty)erect_E[0infty)rightarrowE[0infty)rightarrowE$$ so that $$gamma & # 39;$$ is locally constant for any $$t$$ such that $$gamma (t) in mathrm {int} , E$$ and such that $$gamma & # 39; (t)$$ It becomes your reflection with respect to the hyperplane. $$T _ { gamma (t)} partial E$$ for $$t$$ such that $$gamma (t) in partial E$$.

When $$E$$ it is an ellipse, any path that starts at a focus will return to a focus after a reflection. In one, an ellipsoid 3 is generated by turning the ellipse around the axis that contains the foci, the same property remains true.

Now, imagine that you rotate the ellipse with respect to the other axis of symmetry. Then the image of the pair of spotlights becomes a circle. $$C$$. The question is this: make any path of light from $$C$$ Go back to $$C$$ after a reflection? If this is not true, is this valid for the convex hull of $$C$$?

More generally, can we say anything about ellipsoids, light trajectories and subsets with the property that the paths of light return to them after reflection?

[This was a question made to me by an experimental physicist, but after thinking about it I have been unable to find any reference about it despite its elementary looking form].