Duke Mathematical Journal

From symplectic measurements to the Mahler conjecture

Shiri Artstein-Avidan, Roman Karasev, and Yaron Ostrover

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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.

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

First available in Project Euclid: 8 August 2014

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Zentralblatt MATH identifier

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.)


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

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