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Elementary approach to closed billiard trajectories in asymmetric normed spaces


Authors: Arseniy Akopyan, Alexey Balitskiy, Roman Karasev and Anastasia Sharipova
Journal: Proc. Amer. Math. Soc. 144 (2016), 4501-4513
MSC (2010): Primary 52A20, 52A23, 53D35
Published electronically: May 4, 2016
MathSciNet review: 3531197
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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.


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Additional Information

Arseniy Akopyan
Affiliation: Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria – and – Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994
Email: akopjan@gmail.com

Alexey Balitskiy
Affiliation: Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700 – and – Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994
Email: alexey_m39@mail.ru

Roman Karasev
Affiliation: Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700 – and – Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994
Email: r_n_karasev@mail.ru

Anastasia Sharipova
Affiliation: Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700
Email: independsharik@yandex.ru

DOI: https://doi.org/10.1090/proc/13062
Keywords: Billiards, Minkowski norm, Mahler's conjecture
Received by editor(s): April 5, 2015
Received by editor(s) in revised form: October 5, 2015, and December 14, 2015
Published electronically: May 4, 2016
Additional Notes: The first author was supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n$^{∘}$[291734]
The first and third authors were supported by the Dynasty Foundation
The first, second and third authors were supported by the Russian Foundation for Basic Research grant 15-31-20403 (mol_a_ved).
The second and third authors were supported by the Russian Foundation for Basic Research grant 15-01-99563 A
Communicated by: Patricia Hersh
Article copyright: © Copyright 2016 American Mathematical Society