A Santaló-type Inequality for the $\mathcal{J}$ Transform

Abstract

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

19 pages, 2 figures