Billiard Trajectories inside Cones

Recently it was proved that every billiard trajectory inside a $$C^{3}$$ convex cone has a finite number of reflections. Here, by a $$C^{3}$$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex, closed $$C^{3}$$ hypersurface of that hyperplane, with an everywhere nondegenerate second fundamental form. In this paper, we prove that there exist $$C^{2}$$ convex cones with billiard trajectories that undergo infinitely many reflections in finite time. We also provide an estimation of the number of reflections for billiard trajectories inside elliptic cones in $$\mathbb{R}^{3}$$ using two first integrals.