Periodic billiard trajectories in smooth convex bodies

Abstract

We consider billiard trajectories in a smooth convex body in ${\mathbb{R}}^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we obtain the lower bound $(d-2)(p-1)+2$, which is much better than the previous estimates.

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