Email this article If a Minkowski billiard is projective, then it is the standard billiard
- Authors: Glutsyuk A.A.1,2,3, Matveev V.S.4
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Affiliations:
- Higher School of Contemporary Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
- CNRS, UMR 5669 (UMPA, ENS de Lyon), Lyon, France
- National Research University Higher School of Economics, Moscow, Russia
- Institute of Mathematics, Friedrich Schiller University of Jena, Jena, Germany
- Issue: Vol 216, No 5 (2025)
- Pages: 64-82
- Section: Articles
- URL: https://journal-vniispk.ru/0368-8666/article/view/306705
- DOI: https://doi.org/10.4213/sm10182
- ID: 306705
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Abstract
In the recent paper [5] the first-named author proved that if a billiard in a convex domain in $\mathbb R^n$ is simultaneously projective and Minkowski, then it is the standard Euclidean billiard in an appropriate Euclidean structure. The proof was quite complicated and required high smoothness. Here we present a direct simple proof of this result which works in $C^1$-smoothness. In addition, we prove the semi-local and local versions of this result.
About the authors
Alexey Antonovich Glutsyuk
Higher School of Contemporary Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia; CNRS, UMR 5669 (UMPA, ENS de Lyon), Lyon, France; National Research University Higher School of Economics, Moscow, Russia
Author for correspondence.
Email: aglutsyu@ens-lyon.fr
Doctor of physico-mathematical sciences, no status
Vladimir Sergeevich Matveev
Institute of Mathematics, Friedrich Schiller University of Jena, Jena, Germany
Email: vladimir.matveev@uni-jena.de
Candidate of physico-mathematical sciences, Professor
References
- M. Arnold, S. Tabachnikov, “Remarks on Joachimsthal integral and Poritsky property”, Arnold Math. J., 7:3 (2021), 483–491
- Sh. Artstein-Avidan, R. Karasev, Y. Ostrover, “From symplectic measurements to the Mahler conjecture”, Duke Math. J., 163:11 (2014), 2003–2022
- Sh. Artstein-Avidan, Y. Ostrover, “Bounds for Minkowski billiard trajectories in convex bodies”, Int. Math. Res. Not. IMRN, 2014:1 (2014), 165–193
- M. Berger, “Seules les quadriques admettent des caustiques”, Bull. Soc. Math. France, 123:1 (1995), 107–116
- А. А. Глуцюк, “О гамильтоновых проективных бильярдах на границах произведений выпуклых тел”, Труды МИАН, 327, Математические аспекты механики (2024), 44–62
- E. Gutkin, S. Tabachnikov, “Billiards in Finsler and Minkowski geometries”, J. Geom. Phys., 40:3-4 (2002), 277–301
- P. Haim-Kislev, Y. Ostrover, A counterexample to Viterbo's conjecture
- Tianyu Ma, V. S. Matveev, I. Pavlyukevich, “Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds”, J. Geom. Anal., 31:12 (2021), 12446–12484
- V. S. Matveev, “Riemannian metrics having common geodesics with Berwald metrics”, Publ. Math. Debrecen, 74:3-4 (2009), 405–416
- V. S. Matveev, H.-B. Rademacher, M. Troyanov, A. Zeghib, “Finsler conformal Lichnerowicz–Obata conjecture”, Ann. Inst. Fourier (Grenoble), 59:3 (2009), 937–949
- V. S. Matveev, M. Troyanov, “Completeness and incompleteness of the Binet–Legendre metric”, Eur. J. Math., 1:3 (2015), 483–502
- V. S. Matveev, M. Troyanov, “The Binet–Legendre metric in Finsler geometry”, Geom. Topol., 16:4 (2012), 2135–2170
- V. S. Matveev, M. Troyanov, “The Myers–Steenrod theorem for Finsler manifolds of low regularity”, Proc. Amer. Math. Soc., 145:6 (2017), 2699–2712
- S. Tabachnikov, “Introducing projective billiards”, Ergodic Theory Dynam. Systems, 17:4 (1997), 957–976
- C. Viterbo, “Metric and isoperimetric problems in symplectic geometry”, J. Amer. Math. Soc., 13:2 (2000), 411–431
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