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Mahler’s conjecture for some hyperplane sections

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Abstract

We use symplectic techniques to obtain partial results on Mahler’s conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of p-balls or the Hanner polytopes.

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References

  1. A. Akopyan, A. Balitskiy, R. Karasev and A. Sharipova, Elementary approach to closed billiard trajectories in asymmetric normed spaces, Proceedings of the American Mathematical Society 144 (2016), 4501–4513.

    Article  MathSciNet  Google Scholar 

  2. A. Akopyan, A. Hubard and R. Karasev, Lower and upper bounds for the waists of different spaces, Topological Methods in Nonlinear Analysis 53 (2019), 457–490.

    MathSciNet  MATH  Google Scholar 

  3. A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closed curves on the unit spheres, https://arxiv.org/abs/1801.00242.

  4. S. Artstein-Avidan, R. Karasev and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Mathematical Journal 163 (2014), 2003–2022.

    Article  MathSciNet  Google Scholar 

  5. S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, International Mathematics Research Notices 2014 (2014), 165–193.

    Article  MathSciNet  Google Scholar 

  6. A. Balitskiy, Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture, Geometriae Dedicata 184 (2016), 121–134.

    Article  MathSciNet  Google Scholar 

  7. K. Ball, Mahler’s conjecture and wavelets, Discrete and Computational Geometry 13 (1995), 271–277.

    Article  MathSciNet  Google Scholar 

  8. I. Bárány and L. Lovász, Borsuk’s theorem and the number of facets of centrally symmetric polytopes, Acta Mathematica Hungarica 40 (1982), 323–329.

    MathSciNet  MATH  Google Scholar 

  9. A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, Vol. 1764, Springer, Berlin, 2001.

    MATH  Google Scholar 

  10. F. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proceedings of the American Mathematical Society 76 (1979), 186–188.

    MathSciNet  MATH  Google Scholar 

  11. I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Mathematische Zeitschrift 200 (1989), 355–378.

    Article  MathSciNet  Google Scholar 

  12. M. Fradelizi, A. Hubard, M. Meyer, E. Roldán-Pensado and A. Zvavitch, Equipartitions and Mahler volumes of symmetric convex bodies, http://arxiv.org/abs/1904.10765.

  13. M. Gromov, Isoperimetry of waists and concentration of maps, Geometric and Functional Analysis 13 (2003), 178–215.

    Article  MathSciNet  Google Scholar 

  14. P. M. Gruber, Convex and Discrete Geometry, Grundlehren der mathematischen Wissenschaften, Vol. 336, Springer, Berlin, 2007.

    MATH  Google Scholar 

  15. H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Modern Birkhäuser Classics, Birkhäuser, Basel, 2011.

    Google Scholar 

  16. H. Iriyeh and M. Shibata, Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case, Duke Mathematical journal 169 (2020), 1077–1134.

    Article  MathSciNet  Google Scholar 

  17. J. Kim, Minimal volume product near Hanner polytopes, Journal of Functional Analysis 266 (2014), 2360–2402.

    Article  MathSciNet  Google Scholar 

  18. G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geometric and Functional Analysis 18 (2008), 870–892.

    Article  MathSciNet  Google Scholar 

  19. J. Latschev, D. McDuff and F. Schlenk, The Gromov width of 4-dimensional tori, Geometry & Topology 17 (2013), 2813–2853.

    Article  MathSciNet  Google Scholar 

  20. M. A. Lopez and S. Reisner, A special case of Mahler’s conjecture, Discrete and Computational Geometry 20 (1998), 163–177.

    Article  MathSciNet  Google Scholar 

  21. K. Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis pro Pĕstování Matematiky a Fysiky 68 (1939), 93–102.

    Article  Google Scholar 

  22. M. Meyer and S. Reisner, Shadow systems and volumes of polar convex bodies, Mathematika 53 (2006), 129–148.

    Article  MathSciNet  Google Scholar 

  23. F. Nazarov, F. Petrov, D. Ryabogin and A. Zvavitch, A remark on the Mahler conjecture: Local minimality of the unit cube, Duke Mathematical Journal 154 (2010), 419–430.

    Article  MathSciNet  Google Scholar 

  24. Y. Ostrover, When symplectic topology meets Banach space geometry, in Proceedings of the International Congress of Mathematicians—Seoul 2014, Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 959–981.

    Google Scholar 

  25. F. Schlenk, Embedding Problems in Symplectic Geometry, De Gruyter Expositions in Mathematics, Vol. 40, Walter de Gruyter, Berlin, 2005.

    Book  Google Scholar 

  26. C. Viterbo, Metric and isoperimetric problems in symplectic geometry, Journal of the American Mathematical Society 13 (2000), 411–431.

    Article  MathSciNet  Google Scholar 

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Correspondence to Roman Karasev.

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Supported by the Federal professorship program grant 1.456.2016/1.4, by the Russian Foundation for Basic Research grants 18-01-00036 and 19-01-00169.

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Karasev, R. Mahler’s conjecture for some hyperplane sections. Isr. J. Math. 241, 795–815 (2021). https://doi.org/10.1007/s11856-021-2114-4

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  • DOI: https://doi.org/10.1007/s11856-021-2114-4

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