  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 $\mathbb{R}^n$.
