Skip to main content
SpringerLink
Log in

A Brunn–Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

  • Published:
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Aims and scope Submit manuscript

Abstract

In this paper, we firstly generalize the Brunn–Minkowski type inequality for Ekeland–Hofer–Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan–Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan–Ostrover in 2012.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Non applicable.

Notes

  1. The preprint was split into two papers, which were submitted independently. The present paper is one of them, mainly consisting of contents in Sections 8, 9 of [13].

References

  1. Albers, P., Mazzucchelli, M.: Periodic bounce orbits of prescribed energy. Int. Math. Res. Not. IMRN 14, 3289–3314 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Artstein-Avidan, S., Ostrover, Y.: A Brunn–Minkowski inequality for symplectic capacities of convex domains. Int. Math. Res. Not. IMRN 13, Art. ID rnn044 (2008)

  3. Artstein-Avidan, S., Ostrover, Y.: Bounds for Minkowski billiard trajectories in convex bodies. Int. Math. Res. Not. IMRN 1, 165–193 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Artstein-Avidan, S., Karasev, R., Ostrover, Y.: From symplectic measurements to the Mahler conjecture. Duke Math. J. 163(11), 2003–2022 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clarke, F.H.: Optimization and nonsmooth analysis. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. Wiley, New York (1983)

  6. Ekeland, I.: Convexity methods in Hamiltonian mechanics. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19. Springer, Berlin (1990)

  7. Ekeland, I., Hofer, H.: Symplectic topology and Hamiltonian dynamics. Math. Z. 200, 355–378 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ekeland, I., Hofer, H.: Symplectic topology and Hamiltonian dynamics II. Math. Z. 203, 553–567 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ghomi, M.: Shortest periodic billiard trajectories in convex bodies. Geom. Funct. Anal. 14(2), 295–302 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hofer, H., Zehnder, E.: A new capacity for symplectic manifolds. In: Analysis et cetera, pp. 405–429 (1990)

  11. Irie, K.: Symplectic capacity and short periodic billiard trajectory. Math. Z 272(3–4), 1291–1320 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Irie, K.: Periodic billiard trajectories and Morse theory on loop spaces. Comment. Math. Helv 90(1), 225–254 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jin, R., Lu, G.: Generalizations of Ekeland–Hofer and Hofer–Zehnder symplectic capacities and applications. arXiv:1903.01116v2 [math.SG], 15 May (2019)

  14. Künzle, A.F.: Singular Hamiltonian systems and symplectic capacities. In: Singularities and Differential Equations (Warsaw, 1993), pp. 171–187. Banach Center Publications 33. Polish Academy of Sciences, Warsaw (1996)

  15. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  16. Schneider, R.: Stability for some extremal properties of the simplex. J. Geom. 96(1–2), 135–148 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sikorav, J.-C.: Systémes Hamiltoniens et topologie symplectique. Dipartimento di Matematica dell’Universitá di Pisa, 1990. ETS, EDITRICE PISA

  18. Viterbo, C.: Metric and isoperimetic problems in symplectic geometry. J. Am. Math. Soc. 13(2), 411–431 (2000)

    Article  MATH  Google Scholar 

  19. Yang, F., Wei, Z.: Generalized Euler identity for subdifferentials of homogeneous functions and applications. J. Math. Anal. Appl. 337(1), 516–523 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are deeply grateful to the anonymous referees for giving very helpful comments and suggestions to improve the exposition.

Author information

Authors and Affiliations

  1. Department of Mathematics, Civil Aviation University of China, Tianjin, 300300, People’s Republic of China

    Rongrong Jin

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Guangcun Lu

Authors
  1. Rongrong Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangcun Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangcun Lu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Janko Latschev.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Partially supported by the NNSF 11271044 of China

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jin, R., Lu, G. A Brunn–Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories. Abh. Math. Semin. Univ. Hambg. 93, 1–30 (2023). https://doi.org/10.1007/s12188-023-00263-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12188-023-00263-z

Keywords

Mathematics Subject Classification

Search

Navigation