  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.
