  This paper deals with an analog of the Mahler volume product related to the
${\cal J}$ transform acting in the class of geometric convex functions
${\rm{Cvx}}_0({\mathbb R}^n)$. We provide asymptotically sharp bounds for the
quantity $s^{\cal J}(f) = \frac{\int e^{-{\cal J} f}}{\int e^{-f}}$ and
characterize all the extremal functions.
