Shortest closed billiard orbits on convex tables

TL;DR

This paper presents an algorithm to find the shortest generalized closed billiard orbits on convex tables, including polygons and smooth boundaries, with applications to symplectic capacity computations.

Contribution

It introduces a finite algorithm for polygons and an approximation scheme for general convex tables, extending previous work to Minkowski billiards and symplectic geometry.

Findings

Shortest orbit in regular n-gons is 2-bounce, twice the width.

Algorithm computes Ekeland-Hofer-Zehnder capacity for specific domains.

Method applies to Minkowski billiards and uses Fagnano triangle uniqueness.

Abstract

Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is finite if $T$ is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular $n$-gon $R_n$ is 2-bounce for $n \ge 4$, with length twice the width of $R_n$. As an application we obtain an algorithm computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domain $T \times B^2$ in the standard symplectic vector space $\mathbb{R}^4$. Our method is based on the work of Bezdek-Bezdek and on the uniqueness of the Fagnano triangle in acute triangles. It works, more generally, for planar Minkowski billiards.