Shortest billiard trajectories

TL;DR

This paper proves that convex bodies in Euclidean space have shortest billiard trajectories of bounded period, and specifically in plane fat disk-polygons, these are always 2-periodic, addressing a recent open question.

Contribution

The paper establishes bounds on the period of shortest billiard trajectories in convex bodies and characterizes 2-periodic trajectories in fat disk-polygons in the plane.

Findings

Shortest billiard trajectories in convex bodies have period at most d+1.

In fat disk-polygons, shortest trajectories are always 2-periodic.

Provides partial answers to Zelditch's question on convex bodies with 2-periodic shortest trajectories.

Abstract

In this paper we prove that any convex body of the d-dimensional Euclidean space (d>1) possesses at least one shortest generalized billiard trajectory moreover, any of its shortest generalized billiard trajectories is of period at most d+1. Actually, in the Euclidean plane we improve this theorem as follows. A disk-polygon with parameter r>0 is simply the intersection of finitely many (closed) circular disks of radii r, called generating disks, having some interior point in common in the Euclidean plane. Also, we say that a disk-polygon with parameter r>0 is a fat disk-polygon if the pairwise distances between the centers of its generating disks are at most r. We prove that any of the shortest generalized billiard trajectories of an arbitrary fat disk-polygon is a 2-periodic one. Also, we give a proof of the analogue result for {\epsilon}-rounded disk-polygons obtained from fat disk-polygons by rounding them off using circular disks of radii {\epsilon}>0. Our theorems give partial answers to the very recent question raised by S. Zelditch on characterizing convex bodies whose shortest periodic billiard trajectories are of period 2.