A Santal\'{o}-type Inequality for the ${\cal J}$ Transform

TL;DR

This paper establishes sharp bounds for a Mahler-type volume product involving the ${ m J}$ transform on convex functions, extending classical inequalities to a functional setting and characterizing extremal functions.

Contribution

It introduces a new inequality for the ${ m J}$ transform on convex functions and characterizes the extremal functions achieving equality.

Findings

Derived asymptotically sharp bounds for $s^{ m J}(f)$

Characterized all extremal functions for the inequality

Extended Mahler volume concepts to the ${ m J}$ transform setting

Abstract

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.