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.