Shortest closed billiard orbits on convex tables

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 (Geom. Dedicata 141:197–206, 2009) and on the uniqueness of the Fagnano triangle in acute triangles. It works, more generally, for planar Minkowski billiards.