A note on Santal\'{o} inequality for the polarity transform and its reverse

TL;DR

This paper establishes sharp Santaló and reverse Santaló inequalities for the polarity transform within the class of even geometric log-concave functions, extending fundamental inequalities in convex analysis.

Contribution

It proves the first sharp Santaló and reverse inequalities for the polarity transform in the setting of geometric log-concave functions, a recent area of interest.

Findings

Established sharp bounds for Santaló inequality.

Proved reverse Santaló inequality with optimal constants.

Extended classical inequalities to a new functional setting.

Abstract

We prove a Santal\'{o} and a reverse Santal\'{o} inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1 at the origin, also called geometric log-cancave functions. The bounds are sharp up to the optimal universal constants.