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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
DOI: https://doi.org/10.1090/proc/13062
Published electronically: May 4, 2016
MathSciNet review: 3531197
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] J. C. Álvarez Paiva and F. Balacheff, Contact geometry and isosystolic inequalities, Geom. Funct. Anal. 24 (2014), no. 2, 648-669. MR 3192037, https://doi.org/10.1007/s00039-014-0250-2
  • [2] J.-C. Álvarez Paiva, F. Balacheff, and K. Tzanev,
    Isosystolic inequalities for optical hypersurfaces,
    2013,
    arXiv:1308.5522.
  • [3] Shiri Artstein-Avidan, Roman Karasev, and Yaron Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J. 163 (2014), no. 11, 2003-2022. MR 3263026, https://doi.org/10.1215/00127094-2794999
  • [4] Shiri Artstein-Avidan and Yaron Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not. IMRN 1 (2014), 165-193. MR 3158530
  • [5] Dániel Bezdek and Károly Bezdek, Shortest billiard trajectories, Geom. Dedicata 141 (2009), 197-206. MR 2520072 (2010h:52009), https://doi.org/10.1007/s10711-009-9353-6
  • [6] K. Bezdek and R. Connelly, Covering curves by translates of a convex set, Amer. Math. Monthly 96 (1989), no. 9, 789-806. MR 1033346 (90k:52020), https://doi.org/10.2307/2324841
  • [7] Greg Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870-892. MR 2438998 (2009i:52005), https://doi.org/10.1007/s00039-008-0669-4
  • [8] Kurt Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys. 68 (1939), 93-102 (German). MR 0001242 (1,202c)
  • [9] Serge Tabachnikov, Geometry and billiards, Student Mathematical Library, vol. 30, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. MR 2168892 (2006h:51001)
  • [10] T. Tao,
    Open question: The Mahler conjecture on convex bodies,
    2007,
    terrytao.wordpress.com.

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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

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