From symplectic measurements to the Mahler conjecture

By Shiri Artstein-Avidan, Roman N. Karasev, Y. Ostrover. 2013

108 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7be97e18a620625818e2c0d90794867e31a7833a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Billiards in ellipses revisited (2020, background): “–Zehnder capacity, which made it possible to apply symplectic geometry to convex geometry problems. In particular it was shown that the famous Mahler conjecture followed from the Viterbo’s conjecture [2, 1]. 4. AnotherProof of the FirstResult As before, let u1,...,u n be the unit vectors along the s”
Symplectic embedding problems, old and new (2017, background): “1); Fibonacci and Pell numbers with ratios converging to the Golden and Silver Means [42, 84]; elementary but intricate combinatorial problems and discrete isoperimetric inequalities [62,64]; relations to the lengths of closed billiard orbits [45]; Fourier–Dedekind sums and new examples of lattice p”
Sharp systolic inequalities for Reeb flows on the three-sphere (2015, background): “The validity of this conjecture would have far reaching consequences, as it would imply for instance the Mahler conjecture in convex geometry, see [AAKO13].”