Regularity results for shortest billiard trajectories in convex bodies in $\mathbb{R}^n$

We derive properties of closed billiard trajectories in convex bodies in $\mathbb{R}^n$. Building on techniques introduced by K. and D. Bezdek we establish two regularity results for length minimizing closed billiard trajectories: one for billiard trajectories in general convex bodies, the other for billiard trajectories in the special case of acute convex polytopes. Moreover, we attach particular importance to various examples, also including examples which show the sharpness of the first regularity result. Finally, we show how our results can be used in order to calculate (analytically and by computer) length minimizing closed regular billiard trajectories in convex polytopes.