2014/08/22 by Naeem Alkoumi, Alkoumi, Naeem, Felix Schlenk +1
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.DS #math.SG
paper · pdf · doi:10.48550/arxiv.1408.5255
16 pages, 11 figures
arxiv created 2014/08/22 · arxiv updated 2014/08/25
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 Rn is 2-bounce for n ≥ 4, with length twice the width of Rn. As an application we obtain an algorithm computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domain T × B2 in the standard symplectic vector space ℝ4. 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.