Fulltext

Shortest closed billiard orbits on convex tables

Alkoumi, Naeem ; Schlenk, Felix

In: Manuscripta Mathematica, 2015, vol. 147, no. 3-4, p. 365-380

Add to personal list
    Title
    • Shortest closed billiard orbits on convex tables
    Author
    • Alkoumi, Naeem. Mathematics Department, Birzeit University, PO Box 14, Birzeit, Ramallah, Palestine
    • Schlenk, Felix. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland
    Document Type
    • Postprint
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:331689
    Summary
    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 $${n \ge 4}$$ n ≥ 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 $${T \times B^2}$$ T × B 2 in the standard symplectic vector space $${{\mathbb{R}}^4}$$ 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.