Abstract: We study Birkhoff billiards inside cones in $\mathbb{R}^n$. We show that every trajectory inside a cone over a $C^3$ strictly convex closed hypersurface embedded in $\mathbb{R}^{n-1}$ with non-degenerate second fundamental form undergoes only finitely many reflections. Using this result, we prove that the system is both superintegrable and completely integrable. To our knowledge, this provides the first example of an integrable billiard in which the billiard table is neither a quadric nor composed of pieces of quadrics. The results are obtained with Siyao Yin.
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