From Symplectic Measurements to the Mahler Conjecture

2013/03/31 by Shiri Artstein-Avidan, Roman Karasev, Yaron Ostrover · 2 citations
Mathematics · #math.MG #math.SG #msc:37D50 #msc:52A20 #msc:52A23 #msc:52A40 #msc:53D35

paper · pdf · doi:10.1215/00127094-2794999

published as Duke Math. J. 163, no. 11 (2014), 2003-2022 · 17 pages, revised version. A generalization of Theorem 1.7 has been added (see Remark 4.2)

arxiv created 2013/10/31 · arxiv updated 2015/01/14

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.

Cited by

• Symplectic capacities of S¹-invariant dynamically convex domains in ℝ⁴
• Open problems in billiards and quantitative symplectic geometry

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