Shortest periodic billiard trajectories in convex bodies

Abstract

We show that the length of any periodic billiard trajectory in any convex body $K \subset \mathbf{R}^n$ is always at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, e.g., K is centrally symmetric, in which case we prove that the shortest periodic trajectories are all bouncing ball (2-link) orbits.