A regularity result for shortest generalized billiard trajectories in convex bodies in mathbb {R}^n

Abstract

We study length-minimizing closed generalized Euclidean billiard trajectories in convex bodies in mathbb {R}^n and investigate their relation to the inclusion minimal affine sections that contain these trajectories. We show that when passing to these sections, the length-minimizing closed billiard trajectories are still billiard trajectories, but their length-minimality as well as their regularity can be destroyed. In light of this, we prove what weaker regularity is actually preserved under passing to these sections. Based on the results, we develop an algorithm in order to calculate length-minimizing closed regular billiard trajectories in convex polytopes in mathbb {R}^n.