From Symplectic Measurements to the Mahler Conjecture
Metric Geometry
2015-01-14 v2 Symplectic Geometry
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.
Keywords
Cite
@article{arxiv.1303.4197,
title = {From Symplectic Measurements to the Mahler Conjecture},
author = {Shiri Artstein-Avidan and Roman Karasev and Yaron Ostrover},
journal= {arXiv preprint arXiv:1303.4197},
year = {2015}
}
Comments
17 pages, revised version. A generalization of Theorem 1.7 has been added (see Remark 4.2)