Elementary approach to closed billiard trajectories in asymmetric normed spaces

Abstract

We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.

arXiv comment10 figures added. The title changed

References in corpus (3)

• From Symplectic Measurements to the Mahler Conjecture
• Isosystolic inequalities for optical hypersurfaces
• Contact geometry and isosystolic inequalities

Cited by in corpus (9)

• When symplectic topology meets Banach space geometry
• Bang's problem and symplectic invariants
• Mahler's conjecture for some hyperplane sections
• Equality cases in Viterbo's conjecture and isoperimetric billiard inequalities
• Shortest Closed Billiard Trajectories in the Plane and Equality Cases in Mahler's Conjecture
• Estimating symplectic capacities from lengths of closed curves on the unit spheres
• Viterbo's conjecture for certain Hamiltonians in classical mechanics
• When different norms lead to same billiard trajectories?
• Periodic billiard trajectories and Morse theory on loop spaces