Shortest Closed Billiard Trajectories in the Plane and Equality Cases in   Mahler's Conjecture
  In this note we prove some Rogers-Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture. 
Alexey Balitskiy
math.MG