Duke Mathematical Journal

From symplectic measurements to the Mahler conjecture

Shiri Artstein-Avidan, Roman Karasev, and Yaron Ostrover

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains and Mahler’s conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer–Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

Article information

Source
Duke Math. J., Volume 163, Number 11 (2014), 2003-2022.

Dates
First available in Project Euclid: 8 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1407501717

Digital Object Identifier
doi:10.1215/00127094-2794999

Mathematical Reviews number (MathSciNet)
MR3263026

Zentralblatt MATH identifier
1330.52004

Subjects
Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
Secondary: 52A23: Asymptotic theory of convex bodies [See also 46B06] 52A40: Inequalities and extremum problems 37D50: Hyperbolic systems with singularities (billiards, etc.)

Citation

Artstein-Avidan, Shiri; Karasev, Roman; Ostrover, Yaron. From symplectic measurements to the Mahler conjecture. Duke Math. J. 163 (2014), no. 11, 2003--2022. doi:10.1215/00127094-2794999. https://projecteuclid.org/euclid.dmj/1407501717


Export citation

References

  • [1] S. Artstein-Avidan, V. Milman, and Y. Ostrover, The M-ellipsoid, symplectic capacities and volume, Comment. Math. Helv. 83 (2008), 359–369.
  • [2] S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not. (IMRN) 2012.
  • [3] W. Blaschke, Über affine Geometrie VII: Neue Extremeigenschaften von Ellipse und Ellipsoid, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 69 (1917), 306–318, Ges. Werke 3, 246–258.
  • [4] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $\mathbb{R}^{n}$, Invent. Math. 88 (1987), 319–340.
  • [5] K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk, “Quantitative symplectic geometry” in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ. 54, Cambridge Univ. Press, Cambridge, 2007, 1–44.
  • [6] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics I, Math. Z. 200 (1989), 355–378; II, Math. Z. 203 (1990), 553–567.
  • [7] U. Frauenfelder, V. Ginzburg, and F. Schlenk, “Energy capacity inequalities via an action selector” in Geometry, Spectral Theory, Groups, and Dynamics, Contemp. Math. 387, Amer. Math. Soc., Providence, 2005, 129–152.
  • [8] A. Giannopoulos, G. Paouris, and B. Vritsiou, The isotropic position and the reverse Santaló inequality, to appear in Israel J. Math.
  • [9] Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume product — a new proof, Proc. Amer. Math. Soc. 104 (1988), 273–276.
  • [10] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
  • [11] E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys. 40 (2002), 277–301.
  • [12] D. Hermann, Non-equivalence of symplectic capacities for open sets with restricted contact type boundary, preprint, 1998.
  • [13] H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 25–38.
  • [14] H. Hofer, “Symplectic capacities” in Geometry of Low-Dimensional Manifolds 2 (Durham, 1989), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press, Cambridge, 1990, 15–34.
  • [15] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, Basel, 1994.
  • [16] M. Hutchings, Quantitative embedded contact homology, J. Differential Geom. 88 (2011), 231–266.
  • [17] G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), 870–892.
  • [18] K. Mahler, Ein Übertragungsprinzip für konvexe Korper, Casopis Pyest. Mat. Fys. 68 (1939), 93–102.
  • [19] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Math. Monogr., Clarendon Press, Oxford Univ. Press, New York, 1998.
  • [20] M. Meyer, Une caractérisation volumique de certains espaces normés de dimension finie, Israel J. Math. 55 (1986), 317–326.
  • [21] F. Nazarov, “The Hörmander proof of the Bourgain–Milman theorem” in Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2050, Springer, Heidelberg, 2012, 335–343.
  • [22] F. Nazarov, F. Petrov, D. Ryabogin, and A. Zvavitch, A remark on the Mahler conjecture: Local minimality of the unit cube, Duke Math. J. 154 (2010), 419–430.
  • [23] Y.-G. Oh, Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group, Asian J. Math. 6 (2002), 579–624.
  • [24] S. Reisner, Zonoids with minimal volume product, Math. Z. 192 (1986), 339–346.
  • [25] S. Reisner, Minimal volume-product in Banach spaces with a $1$-unconditional basis, J. London Math. Soc. 36 (1987), 126–136.
  • [26] J. Saint Raymond, “Sur le volume des corps convexes symétriques” in Initiation Seminar on Analysis: G. Choquet–M. Rogalski–J. Saint-Raymond, 20th Year: 1980/1981, Publ. Math. Univ. Pierre et Marie Curie 46, Univ. Paris VI, Paris, 1981, exp. no. 11.
  • [27] L. A. Santaló, Un invariante afin para los cuerpos convexos de espacio de $n$ dimensiones, Port. Math. 8 (1949), 155–161.
  • [28] C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685–710.
  • [29] C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc. 13 (2000), 411–431.