From Symplectic Measurements to the Mahler Conjecture

TL;DR

This paper explores the connection between symplectic geometry and convex geometry by linking a symplectic isoperimetric inequality to Mahler's conjecture, suggesting that maximizing Hofer-Zehnder capacity relates to minimizing volume product.

Contribution

It establishes a novel link between symplectic capacities and Mahler's conjecture, proposing that extremal symplectic properties imply solutions to a longstanding convex geometry problem.

Findings

Maximizing Hofer-Zehnder capacity for convex bodies implies the hypercube minimizes volume product.

The paper connects symplectic isoperimetric inequalities to Mahler's conjecture.

A conditional statement links symplectic and convex geometric conjectures.

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.