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.