Shortest Minkowski billiard trajectories on convex bodies

TL;DR

This paper studies the properties of shortest closed billiard paths in convex bodies under Minkowski/Finsler metrics, providing theoretical insights and an algorithm for planar cases.

Contribution

It establishes regularity and geometric properties of Minkowski billiard trajectories and introduces an algorithm for computing shortest paths in the plane.

Findings

Regularity of length-minimizing trajectories

Geometric characterization of trajectories

Algorithm for planar Minkowski billiards

Abstract

We rigorously investigate closed Minkowski/Finsler billiard trajectories on $n$-dimensional convex bodies. We outline the central properties in comparison and differentiation from the Euclidean special case and establish two main results for length-minimizing closed Minkowski/Finsler billiard trajectories: one is a regularity result, the other is of geometric nature. Building on these results, we develop an algorithm for computing length-minimizing closed Minkowski/Finsler billiard trajectories in the plane.