The Scaled Polarity transform and related inequalities

TL;DR

This paper introduces scaled polarity transforms for log-concave functions, establishing bounds on Mahler-type products and extending duality results, thus advancing the understanding of functional inequalities in convex analysis.

Contribution

It presents a rescaling of the polarity transform to normalize Mahler product bounds and extends duality results to geometric log-concave functions.

Findings

Rescaled polarity transform achieves asymptotically tight Mahler product bounds.

Establishes a similar result for the $\

Extends K"onig-Milman duality of entropy to geometric log-concave functions.

Abstract

In this paper we deal with generalizations of the Mahler volume product for log-concave functions. We show that the polarity transform $\mathcal A$ can be rescaled so that the Mahler product it induces has upper and lower bounds of the same asymptotics. We discuss a similar result for the $\mathcal J$ transform. As an application, we extend the K\"onig-Milman duality of entropy result to the class of geometric log-concave functions.